Fullduplex distributed switchandstay networks with wireless energy harvesting: design and outage analysis
 Quang Nhat Le^{1},
 Nhu Tri Do^{1},
 Vo Nguyen Quoc Bao^{2} and
 Beongku An^{3}Email authorView ORCID ID profile
https://doi.org/10.1186/s1363801607840
© The Author(s) 2016
Received: 28 June 2016
Accepted: 28 November 2016
Published: 15 December 2016
Abstract
Distributed switch and stay combining (DSSC) has been considered as an effective technique to achieve spatial diversity in a distributed manner with low processing complexity. In this paper, we incorporate DSSC into a fullduplex dual hop relaying system, where two energyconstrained relay nodes assist the information transmission from the source to the destination using energy wirelessly harvested from the source. By applying DSSC at the destination, this efficient technique not only improves performance of fullduplex relaying systems and increases the system diversity but also reduces the implementation and hardware complexity at the destination. We obtain tight approximate expression for outage probability in the case of using decodeandforward (DF) relaying protocol and analytical expression for outage probability in the case of using amplifyandforward (AF) relaying protocol. Numerical results show that our DSSC fullduplex relaying with wireless information and power transfer (WIPT) system achieves the full spatial diversity and has a better performance in terms of outage probability than that of the DSSC halfduplex relaying and conventional fullduplex relaying with WIPT system.
Keywords
1 Introduction
Wireless energy transfer through radio frequency (RF) signals has emerged as the promising and effective solution to supply power and prolong the lifetime of energyconstrained wireless networks. Since RF signals can carry both information and energy, it has recently attracted a lot of attention on research activities, especially on investigating simultaneously wireless information and power transfer (SWIPT) [1–3]. In [4], an energy constrained relay node using amplifyandforward (AF) protocol was considered based on the time switching and power splitting receiver architectures, respectively. Specifically, the authors investigated both the delaylimited and the nondelaylimited transmission and derived the analytical expressions for the outage probability and ergodic capacity, respectively. Later on, the same authors extended the analysis to the throughput and ergodic capacity of DF relaying systems in [5]. Based on the time switching receiver architecture, the authors in [6] proposed adaptive time switching protocols for RF energy harvesting and information transmission for both AF and DF networks. They obtained analytical expressions of the achievable throughput for both the continuous and the discrete time switching protocols. It is pointing out that these authors only consider halfduplex relaying mechanism, where relay node cannot receive and transmit data simultaneously in the same frequency band.
With the advance of antenna and signal processing technology, fullduplex (FD) has received significant attention, where wireless nodes can receive and transmit information simultaneously in the same frequency band (see references [7–10] and therein). In [11], the authors investigated harvestuse (HU) relaying system with halfduplex and fullduplex operation mode. The main conclusion in [11] is that FD seems to be an attractive and promising technology for HUbased cooperative networks. In [12], the authors considered a time switchingbased fullduplex wireless powered relaying system, where the relay node operated in fullduplex mode with simultaneously information reception and information transmission. Specifically, they provided an analytical characterization of the achievable throughput of three different communication modes, namely, instantaneous transmission, delayconstrained transmission, and delay tolerant transmission by optimizing time splitting ratio. In [13], the authors examined a beamforming scheme in a time switchingbased relaying energy harvesting system with multiantenna fullduplex DF relays. Moreover, a fullduplex wirelesspowered transmission system with simultaneous information reception and energy transmission was investigated in [14]. In [15], the authors proposed a novel twophase protocol for an AF relay to activate fullduplex transmission. Especially, they regarded the energy from the loopback—interference channel as useful energy and integrated this energy into the wirelesspowered transfer process, and the optimal power allocation and beamforming design at the relay node were derived. However, the performance and diversity of fullduplex relaying with WIPT networks can be further improved.

We proposed the application of distributed switch and stay combining (DSSC) to fullduplex relaying with WIPT networks to improve system performance. This is the first work of considering this combination.

We derived the performance of our proposed system in terms of the outage probability (OP) over Rayleigh fading channels for both two typical relaying protocols, AF and DF protocols. Through some simulation results, it is shown that our proposed protocol can attain the full spatial diversity if the preselected switching threshold is chosen equal or higher than the outage threshold. Otherwise, the diversity of one is obtained. This is verified by numerical results and diversity analysis.

We showed that our proposed system has the better performance than the DSSC halfduplex relaying system and conventional fullduplex relaying one according to the system outage probability obtained by numerical results.
The rest of the paper is organized as follows. In Section 2, DSSC fullduplex relaying with WIPT systems is presented. In Section 3, we formulate the outage probability of our proposed system in two different cases: two relays operate in DF mode and two relays operate in AF mode. Section 4 provides diversity analysis of DF and AF cases. In Section 5, we consider halfduplex relaying using DSSC which serves as a benchmark. In Section 6, numerical results are presented. Finally, Section 7 concludes the paper.
2 System model
2.1 System description
All wireless links exhibit fading and addictive white Gaussian noise (AWGN) [11]. The channels are modelled to be frequencyflat, independent, and identically distributed Rayleigh block fading. This means that the channel coefficients are constant over a single block time T, but vary independently from block to block. Let \(h_{\mathrm {SR_{1}}}, h_{\mathrm {R_{1}D}}, h_{\mathrm {SR_{2}}}, h_{\mathrm {R_{2}D}}, f_{\mathrm {R_{1}}}\), and \(f_{\mathrm {R_{2}}}\) be the channel coefficients from the source to relay R _{1}, relay R _{1} to the destination, the source to relay R _{2}, relay R _{2} to the destination, the loopback interference channel of relay R _{1} and the loopback interference channel of relay R _{2}, respectively. Due to Rayleigh fading, the channel powers, denoted by \(h_{\mathrm {SR_{1}}}^{2}, h_{\mathrm {R_{1}D}}^{2}, h_{\mathrm {SR_{2}}}^{2}, h_{\mathrm {R_{2}D}}^{2}, f_{\mathrm {R_{1}}}^{2}\), and \(f_{\mathrm {R_{2}}}^{2}\), are independent and exponential random variables, whose means are \(\lambda _{h_{\mathrm {SR_{1}}}}, \lambda _{h_{\mathrm {R_{1}D}}}, \lambda _{h_{\mathrm {SR_{2}}}}, \lambda _{h_{\mathrm {R_{2}D}}}, \lambda _{f_{\mathrm {R_{1}}}}\), and \(\lambda _{f_{\mathrm {R_{2}}}}\), respectively. Further, the SNR of two branches S→R _{1}→D and S→R _{2}→D are represented by \(\gamma _{\mathrm {R_{1}}}\) and \(\gamma _{\mathrm {R_{2}}}\), respectively.
In this paper, information is transmitted between the source and the destination through either R _{1} (S→ R _{1}→ D) or R _{2} (S→R _{2}→D) link. Therefore, some technique need to be applied at the destination to combine two signals from two links or select one of the twos from two links. To increase the system diversity and also to reduce the complexity at the destination, we propose distributed switch and stay combining (DSSC) technique. The operation of this technique in our model is described as follows. In each transmission block time T, only one link (either S→R _{1}→D or S→R _{2}→D) is active. To determine the active link, depending on which link is active in the current block time T, the destination node compares the received SNR (which equals to \(\gamma _{\mathrm {R_{1}}}\) or \(\gamma _{\mathrm {R_{2}}}\)) with the given threshold κ. The switching occurs (the alternative link will be used in the next block time T) when the instantaneous SNR of the currently selected link falls below the threshold κ regardless of the current instantaneous SNR of the alternative link. The switching process will be implemented by an appropriate feedback signal sent from destination to the source and relays through the dedicated feedback channel, indicating a switching on the transmission path. We assume that there exists an errorfree feedback channel between the source and the destination. More specifically, the source signal reaches the destination through either S→R _{1}→D link or S→R _{2}→D link and in each transmission block time T, only one link is active.
2.2 Signal modeling
In this section, we analyze signal modeling in two phases. In each phase, we consider two cases: relay R _{1} is used for information transmission (S→R _{1}→D is active) and relay R _{2} is used for information transmission (S→ R _{2}→D is active).
2.2.1 In the first phase
where \(d_{\mathrm {SR_{1}}}\) is the distance between the source and relay R _{1},m is the path loss exponent, x _{S} is the information symbol with E {x _{S}^{2}} = \({\mathcal {P}}_{\mathrm {S}}\), E {.} denotes the expectation operation, \({\mathcal {P}}_{\mathrm {S}}\) is the source transmit power and \(n_{\mathrm {R_{1}}}\) is the zero mean additive white Gaussian noise (AWGN) with variance N _{0}.
where \(\overline {\gamma }=\frac {{\mathcal {P}}_{\mathrm {S}}}{N_{0}}\) is the average transmit SNR.
2.2.2 In the second phase
In this paper, we consider both AF and DF relaying protocols.
where \({\mathcal {P}}_{\mathrm {R_{1}}}\) is the transmit power of relay R _{1}.
where \(x_{\mathrm {R_{1}}}\) is the transmit signal of relay R _{1} with E\(\{x_{\mathrm {R_{1}}}^{2}\}\) = \({\mathcal {P}}_{\mathrm {R_{1}}}, {\mathcal {P}}_{\mathrm {R_{1}}}\) is the transmit power of relay \(\mathrm {R_{1}}, d_{\mathrm {R_{1}D}}\phantom {\dot {i}\!}\) is the distance between relay R _{1} and the destination and n _{D} is the zero mean AWGN with variance N _{0}.
where \({\mathcal {P}}_{\mathrm {R_{2}}}\) is the transmit power of relay R _{2}.
Concurrently with information transmission by active relay in the second phase, the relay harvests energy from the energy signal which is transmitted from the source. Here, we assume that the relay harvests dedicated energy from the source, but the relay also recycles its own energy from loopback selfinterference channel [15].
where x _{E} is the energy symbol from the source.
where η is a constant and denotes the energy conversion efficiency. Since we consider the selfinterference link as useful link, thus from (16), the selected relay not only harvests energy from the source but it also scavenges energy from the selfinterference link. Note that, we take into consideration particular case that the average energy used for transmission by the selected relay is equal to that being harvested [15]. Further, we ignore the negligible energy harvested from the receiver noise since the energy harvested from the source is much larger than that of the receiver noise [12].
2.3 Mode of operation
for the case of AF.
Note that \(\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}}\) and \(\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {AF}}}\) are the endtoend SNRs of relay R _{1} when relay R _{1} uses DF and AF protocols, respectively.
for the case of AF.
Note that \(\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}\) and \(\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {AF}}}\) are the endtoend SNRs of relay R _{2} when relay R _{2} uses DF and AF protocols, respectively.
The switching process will be implemented by appropriate feedback sent from the destination to the source and relays, indicating a switching on the transmission path.
3 Performance analysis
In this section, we will study the performance analysis of the considered system in terms of outage probability. Depending on AF or DF protocol which is used at relays, we need to study outage probability of the proposed system in a separated manner.
3.1 Two relays operate in DF mode
where \(F_{\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}}}(\kappa)\) and \(F_{\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}}(\kappa)\) are the cumulative distribution functions (CDFs) of \(\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}}\) and \(\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}\) evaluated at the preselected switching threshold κ, respectively.
To calculate (23) and (24), we first need to derive the cumulative distribution functions (CDFs) of \(\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}}\) and \(\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}\) evaluated at the preselected switching threshold κ.
Lemma 1
Proof
See Appendix.
□
By plugging (25) and (26) into (23) and (24), we will obtain the steady state selection probabilities of S→R _{1}→D link and S→R _{2}→D link.
where conditional outage probabilities in (27) can be obtained by replacing γ _{ M } and t in (28) with the suitable values, i.e., \(\gamma _{M}\in \left \{\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}},\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}\right \}\) and t∈{κ,0}. In this paper, we adopt the cumulative distribution function (CDF) of γ _{ M } evaluated at the outage threshold SNR γ _{ th }, which is expressed as γ _{ th }= 2^{ r }1 with r representing the target rate, coincides with the conditional outage probability when M link is selected with M∈{R _{1},R _{2}}, it means that \(F_{\gamma _{M}}(\gamma _{\text {\texttt {th}}})=\Pr \left \{\gamma _{M}\leq \gamma _{\text {\texttt {th}}}(\gamma _{M}>0)\right \}\) and \(F_{\gamma _{M}}(t)=\Pr (\gamma _{M}<t)=1\Pr (\gamma _{M}>t)\). We note that since \(\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}}\) and \(\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}\) are independent, so \(\Pr (\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}\leq \gamma _{\text {\texttt {th}}}\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}}\leq \kappa)=F_{\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}}(\gamma _{\text {\texttt {th}}})\) and \(\Pr (\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}}\leq \gamma _{\text {\texttt {th}}}\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}\leq \kappa)=F_{\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}}}(\gamma _{\text {\texttt {th}}})\).
By plugging conditional outage probabilities (which are calculated by (28)) and state selection probabilities (which are derived in (23) and (24)) into (27), we obtain the system outage probability as in (30) (on top of the next page) with \(F_{\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}}}(\gamma _{\text {\texttt {th}}})\) and \(F_{\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}}(\gamma _{\text {\texttt {th}}})\) can be calculated by replacing κ in (25) and (26) with γ _{ th }.
We can see that the minimum outage probability in (29) is equal to the outage probability of the system which selects the better link between S→R _{1}→D and S→R _{2}→D link for source information transmission.
3.2 Two relays operate in AF mode
where the last results in (31) and (32) are obtained by plugging (16) and (18) into (9) and (14) and using (2) and (4).
Next, we need to derive the cumulative distribution functions (CDFs) of \(\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {AF}}}\) and \(\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {AF}}}\) evaluated at the predesigned switching threshold κ. \(F_{\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {AF}}}}(\kappa)\) is expressed as follows in (33).
Lemma 2
Proof
See Appendix.
□
We can see that the minimum outage probability in (37) is equal to the outage probability of the system which selects the better link between S→R _{1}→D and S→R _{2}→D link for source information transmission.
4 Diversity analysis
4.1 DF case
where ζ _{2}=α _{2}−ξ _{2} C _{ E } χ _{2} and ν _{2}=ξ _{2} χ _{2}
From (44), (46), and (48), we can see that the system full diversity can be achieved with order of two when κ≥γ _{ th }, whereas the diversity order of the system only attains to one when κ<γ _{ th }.
4.2 AF case
Since (34) and (35) involve an integral, which in general do not admit closedform solution, hence, it is not amenable to further processing and diversity order analysis. However, we will conduct diversity order analysis for AF case by using numerical results, which are shown in Fig. 6.
According to [33], the diversity order can be indicated by the slope at high SNR regions of an outage performance curve. As shown in Fig. 6, we can see that the performance curves of two cases γ _{ th }<κ (corresponds to κ=11) and γ _{ th }=κ (corresponds to κ=7) are parallel to the curve that is proportional to \(\frac {1}{\text {SNR}^{2}}\), which provide the full diversity order while the performance curve of the case γ _{ th }>κ (corresponds to κ=3) is parallel to the curve that is proportional to \(\frac {1}{\text {SNR}}\), which results in the diversity order of one.
5 HD relaying
In this section, we consider the existing Power Splittingbased Relaying (PSR) protocol [4], which serves as a benchmark for performance comparison. The purpose of using PSR protocol is that since PSR has a better performance than time switchingbased relaying (TSR) protocol, which is shown in [4]. Some of the results have been derived in [4]; however, we present here to make our work selfcontained.
Note that, unless otherwise stated, all notations in this section have the same meaning as they have in Subsection 2.2.
where the second equality is obtained by using (50) and \(\overline {\gamma } = \frac {{\mathcal {P}}_{\mathrm {S}}}{N_{0}}\).
The system model for the case of using R _{2} is similar to the case of using R _{1}, so we skip this description for brevity.
5.1 DF case
Lemma 3
\(F_{\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}}}(\kappa)\) is given by
where \(\nu _{1} = \frac {\kappa d_{\mathrm {SR_{1}}}^{m}}{\left (1  \rho \right) \overline {\gamma }}\).
Proof
The proof follows the same step used in Lemma 1. Thus we skip the proof for HD using DF for brevity.
Similarly, \(F_{\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}}(\kappa)\) is expressed as
where \(\nu _{2} = \frac {\kappa d_{\mathrm {SR_{2}}}^{m}}{\left (1  \rho \right) \overline {\gamma }}\).
The system outage probability \(\mathrm {P_{out}^{DF}({\gamma _{th}})}\) for HD using DF is calculated using (30), where \(F_{\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}}}(\kappa)\) and \(F_{\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}}(\kappa)\) are derived in (57) and (59), \(F_{\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {DF}}}}(\gamma _{\text {\texttt {th}}})\) and \(F_{\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {DF}}}}(\gamma _{\text {\texttt {th}}})\) can be calculated by replacing κ in (57) and (59) with γ _{ th }. □
5.2 AF case
This case has been studied in [4] and we present the result here for the purpose of completeness.
Lemma 4
where \(\alpha _{3} = \kappa (1  \rho) \overline {\gamma } d_{\mathrm {SR_{1}}}^{m} d_{\mathrm {R_{1}D}}^{m}, \xi _{3} = \kappa d_{\mathrm {SR_{1}}}^{2m} d_{\mathrm {R_{1}D}}^{m}, \chi _{3} = (1 \rho) \rho \eta \overline {\gamma }^{2}\) and \(\nu _{3} = \eta \overline {\gamma } \rho d_{\mathrm {SR_{1}}}^{m} \kappa \).
where \(\alpha _{4} = \kappa (1  \rho) \overline {\gamma } d_{\mathrm {SR_{2}}}^{m} d_{\mathrm {R_{2}D}}^{m}, \xi _{4} = \kappa d_{\mathrm {SR_{2}}}^{2m} d_{\mathrm {R_{2}D}}^{m}, \chi _{4} = (1 \rho) \rho \eta \overline {\gamma }^{2}\) and \(\nu _{4} = \eta \overline {\gamma } \rho d_{\mathrm {SR_{2}}}^{m} \kappa \).
The system outage probability for HD using AF is calculated using ( 36 ), where \(F_{\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {AF}}}}(\kappa)\) and \(F_{\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {AF}}}}(\kappa)\)are shown in ( 61 ) and ( 62 ), \(F_{\gamma _{\mathrm {e2e,R_{1}}}^{\text {\texttt {AF}}}}(\gamma _{\text {\texttt {th}}})\)and \(F_{\gamma _{\mathrm {e2e,R_{2}}}^{\text {\texttt {AF}}}}(\gamma _{\text {\texttt {th}}})\)can be calculated by replacing κin ( 61 ) and ( 62 ) with γ _{ th }.
6 Numerical results
In this section, we present numerical simulation results to verify our analytical expressions derived in the previous section. Unless otherwise stated, we set the source transmission rate as R _{ c }= 3 bps/Hz; hence, the outage threshold is expressed as \(\phantom {\dot {i}\!}\gamma _{\text {\texttt {th}}}=2^{R_{c}}1=7\). The distances \(\phantom {\dot {i}\!}d_{\mathrm {SR_{1}}}, d_{\mathrm {R_{1}D}}, d_{\mathrm {SR_{2}}}\), and \(d_{\mathrm {R_{2}D}}\) are normalized to unit value. The path loss exponent is set to be m=3, while the energy conversion efficiency is set to be η=0.4. The power splitting ratio ρ is set to be ρ=0.6, which is the optimal value in [4]. We also set \(\lambda _{h_{\mathrm {SR_{1}}}}=1, \lambda _{h_{\mathrm {R_{1}D}}}=1, \lambda _{f_{1}}=0.1, \lambda _{h_{\mathrm {SR_{2}}}}=1, \lambda _{h_{\mathrm {R_{2}D}}}=1, \lambda _{f_{2}}=0.1\).
In Fig. 6, we examine the outage probability between DSSC fullduplex and the conventional fullduplex systems with γ _{ th }=7,κ=3,7,11 when the relay uses AF protocol. It can be shown from Fig. 6 that the simulation results are matched with the analytical results, confirming the correctness of our analysis. As can be readily observed, the outage probability of DSSC fullduplex system is much smaller than that of the conventional full duplex, confirming the positive effect of distributed switch and stay combining (DSSC) technique in improving the performance of the fullduplex relaying system. When the transmit SNR increases, the outage probability of both systems decreases. This is intuitive since when the transmit SNR increases, it means that the destination can receive the better quality of the source information signal from the selected relay and the destination can decode the source signal more correctly, hence reduces the outage event at the destination. Besides that, once again we can also see that the outage probability of DSSC fullduplex system is minimum when γ _{ th }=κ=7. Further, the diversity order achieved by our schemes is examined and compared. Recalling the diversity order can be indicated by the slope at high SNR regions of an outage performance curve [33]. This figure shows that the performance curves of two cases γ _{ th }<κ (corresponds to κ=11) and γ _{ th }=κ (corresponds to κ=7) are parallel to the curve that is proportional to \(\frac {1}{\text {SNR}^{2}}\), which provide the full diversity order while the performance curves of the case γ _{ th }>κ (corresponds to κ=3) and the conventional full duplex are parallel to the curve that is proportional to \(\frac {1}{\text {SNR}}\), which results in the diversity order of one. This makes sense when κ≥γ _{th} the system full diversity can be achieved with order of two since a large switching threshold makes the switch mechanism effective and the two branches can be efficiently exploited. In contrast, when κ<γ _{th}, the diversity order of system gradually goes down to one because a small κ makes the branch switching less likely happen so that diversity can not be efficiently used.
7 Conclusions
In this paper, we investigated the performance of DSSC technique in dualhop fullduplex relaying systems with wireless information and power transfer (WIPT). We obtained tight approximate expression and analytical expression of the system outage probability for DF and AF protocols over Rayleigh fading channels, respectively. Compared to the DSSC halfduplex relaying and conventional fullduplex relaying with WIPT system, our results show that DSSC technique can substantially boost the operational performance of fullduplex relaying systems and achieves the full spatial diversity in cases of the preselected switching threshold is chosen equal or higher than the outage threshold.
8 Appendix
8.1 The proof of Lemma 1
where θ can be solved by using integration by parts and after some simple manipulations, we obtain the last result in (70).
Finally, plugging \(\mu =\frac {\kappa d_{\mathrm {SR_{1}}}^{m}}{\overline {\gamma }}\) into (71), we obtain the desired result as in (25).
8.2 The proof of Lemma 2
where \(\int \limits _{0}^{d/c}f_{X}(x)dx+\int \limits _{d/c}^{\infty }f_{X}(x)dx=\int \limits _{0}^{\infty }f_{X}(x)dx=1\) and \(f_{X}(x)=\frac {1}{\lambda _{h_{\mathrm {SR_{1}}}}}e^{\frac {x}{\lambda _{h_{\mathrm {SR_{1}}}}}}\), the last equality in (74) can be achieved.
where K _{1}(.) is the firstorder modified Bessel function of the second kind ([34], Eq. (8.432.1)) and the last equality is obtained by using the formula in ([34], Eq. (3.324.1)).
Next, denoting \(U=f_{\mathrm {R_{1}}}^{2}\) and conditioned on U from (77), we obtain the desired result as in (34).
Declarations
Acknowledgments
The work of Q. N. Le, N. T. Do and B. An was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2016R1D1A1B03934898) and by the Leading Human Resource Training Program of Regional Neo industry Through the National Research Foundation of Korea (NRF) funded by the Ministry of science, ICT and future planning (Grant No. 2016H1D5A1910577). The work of V. N. Q. Bao was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 102.042014.32.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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