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Outage performance of fixedgain and variablegain AF fullduplex relaying in nonidentical Nakagamim fading channels
 Asil Koç^{1},
 İbrahim Altunbaş^{1} and
 Abbas Yongaçoğlu^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s1363801708881
© The Author(s) 2017
 Received: 30 June 2016
 Accepted: 18 May 2017
 Published: 19 June 2017
Abstract
We investigate outage performance of fixedgain and variablegain amplifyandforward (AF) fullduplex (FD) relaying systems under residual loopinterference (LI) in nonidentical Nakagamim fading channels. New exact outage probability expressions are derived analytically in a singleintegral form by using the cumulative distribution function approach. Moreover, tight lowerbound and asymptotic outage probability expressions are derived in closedform for the evaluated systems. It is shown that the fixedgain AFFD relaying outperforms the variablegain AFFD relaying when LI power and/or signaltonoise ratio increase. The analytical results are verified by MonteCarlo simulations.
Keywords
 Fullduplex relaying
 Nakagamim
 Amplify and forward
 Fixedgain
 Variablegain
 Outage analysis
1 Introduction
FullDuplex (FD) communication allows receiving and transmitting the signal at the same time interval using the same frequency band, hence it improves the spectral efficiency and reduces endtoend delay with respect to the conventional halfduplex communication [1]. Additionally, the FD communication improves the network secrecy by means of the mixed signals during the simultaneous transmission [1]. Therefore, it has recently gained great attention as a potential candidate for the next generation wireless communication systems [2]. Although FD communication causes loopinterference (LI) (or selfinterference) at the receiver antenna, the recent developments on antenna technology and signal processing techniques make it feasible by applying LI cancellation techniques, e.g., antenna separation, analog cancellation and digital cancellation [3–5]. The practical implementations in [4] shows that although the LI power can be decreased 74 dB on average, it cannot be completely removed, even with complex cancellation processes [4–6]. Moreover, the experimental studies as in [4] demonstrate that it is convenient to model the residual LI channel with Rician or Nakagamim fading due to the multipath signal propagation and lineofsight (LOS) component between the transmitter and the receiver.
Cooperative communication with relaying has also attracted considerable attention in recent years since it provides coverage extension, capacity, outage, and error performance enhancement [7]. Amplifyandforward (AF) and decodeandforward (DF) are the most popular signal forwarding methods used at the relays [7]. In the AF method, the amplifying process may be based on the partial or full channel state information of the sourcerelay channel, namely fixedgain [8] and variablegain [9] AF relaying, respectively. The conventional cooperative communication systems suffer from spectral efficiency loss due to halfduplex relaying. In order to improve the spectral efficiency, FD communication is utilized at the relays [10]. In the literature, several papers investigate performance of the AFFD relaying. In [11], optimal power allocation (PA) problem and capacity of the variablegain AFFD relaying are investigated. In [12], outage probability of the variablegain AFFD relaying system with direct link between source and destination is investigated in Rayleigh fading channels and shown that the direct link provides diversity gain. The outage probability of the variablegain AFFD relaying system with direct link is derived for Rayleigh fading channels in [13] under processing delay and residual LI. Note that the sourcerelay link is assumed fadingfree in [13]. In [14], optimal PA problem based on statistical and instantaneous channel state information for the variablegain AFFD relaying system with direct link is investigated for Rayleigh fading under total source and relay power constraint. In that paper, minimum outage probability expressions are derived and comparisons with the DFFD relaying are also presented. An optimal PA analysis based on minimizing the pairwise error probability is carried out for the fixedgain AFFD relaying in [15] and shown that unequal PA presents better error performance than those using equal power in the transmitters. In [16], error performance and diversity of uncoded and coded fixedgain AFFD relaying with direct link are investigated. AFFD relaying is applied to the systems with multiple relays (e.g. in [17–19]) and shown that the performance can be enhanced significantly by using all the relays or the selected relay. The outage performance of variablegain AFFD relaying over Nakagamim channel is recently analyzed in [20] and the exact outage probability expression is derived in terms of extended generalized bivariate MeijerG function (EGBMGF). In [20], an approximate outage performance expression is also obtained in closedform by ignoring additive noise in the sourcerelay channel. Besides, the asymptotic outage probability, asymptotic ergodic capacity and ergodic capacity lowerbound expressions are presented. In [21], the outage performance of the fixedgain AFFD relaying is investigated and compared with the variablegain AFFD relaying over Rayleigh fading channels where the exact outage probability expression is obtained in a singleintegral.
In this paper, we derive exact outage probability expressions for both fixedgain and variablegain dualhop AFFD relaying where all channels are modeled as Nakagamim fading channel including the residual LI channel. The outage probability expressions are obtained by using cumulative distribution function (CDF) of the endtoend signaltointerferenceplusnoise ratio (SINR). Our results in the fixedgain and variablegain AFFD relaying are both in a singleintegral form. To the best of our knowledge, there is no outage performance investigation in the literature for the fixedgain AFFD relaying systems over nonidentical Nakagamim fading channels. Also, our exact outage performance result for the variablegain AFFD relaying can be easily calculated numerically in comparison to the exact outage probability expression in terms of EGBMGF given in [20] which is based on double MellinBarnes type integrals. Furthermore, there is no restriction on the m parameter of the relaydestination link in our analyses. We also present tight lowerbound and asymptotic outage probability expressions of the considered systems in closedform. The theoretical results are validated by computer simulations.

All of the channels are modeled as Nakagamim fading channels including the residual LI channel at the relay node which makes the analyses more accurate for practical purposes.

The exact outage probability expressions of the fixedgain AFFD and variablegain AFFD systems are derived in singleintegral forms which can be easily evaluated by using common mathematical package programs.

For both of the considered systems, the exact outage performance analyses are supported with the lowerbound and asymptotic outage probability expressions obtained in closedform.

The advantages of the fixedgain AFFD relaying with respect to the variablegain AFFD relaying in terms of outage performance are presented. We demonstrate that the system has lower error floor when the fixedgain AFFD relaying is deployed.

It is shown that m parameter of the residual LI channel has dominant effect for the performance limitation in the fixedgain AFFD systems. On the other hand, the dominant m parameter of the variablegain AFFD systems belongs to the sourcerelay link channel for the error floor.
The remainder of the paper is organized as follows. In Section 2, the system model for AFFD relaying is introduced and the SINR expressions are derived for the fixedgain and variablegain relays. The exact outage probability expressions of the proposed systems are given in Section 3. Then, the lowerbound and asymptotic outage probability analyses are given in Section 4. The theoretical and simulation results for different scenarios are demonstrated in Section 5. Finally, Section 6 concludes the paper.
2 System model and endtoend SINR
where 〈.〉 represents the time average over one symbol duration, \({\gamma _{SR}}={P_{S}}{h_{SR}}{^{2}}/\sigma _{R}^{2}\), \({\gamma _{RD}} = {P_{R}}{h_{RD}}{^{2}}/\sigma _{D}^{2}\) and \({\gamma _{LI}} = {P_{R}}{h_{LI}}{^{2}}/\sigma _{R}^{2}\). By defining \({\bar \gamma _{SR}} = {P_{S}}{\Omega _{SR}}/\sigma _{R}^{2}\), \({\bar \gamma _{RD}} = {P_{R}}{\Omega _{RD}}/\sigma _{D}^{2}\), \({\bar \gamma _{LI}} = {P_{R}}{\Omega _{LI}}/\sigma _{R}^{2}\) and Ω _{ A }=E[ h _{ A }^{2}] with A∈{SR,RD,LI}, probability density function (PDF) and CDF of these random variables can be given, respectively, by \({p_{{\gamma _{A}}}}(x) = {{\mu _{A}^{m_{A}}}{x^{{m_{A}}  1}} \over {\Gamma ({m_{A}})}}{e^{ x{\mu _{A}}}}\) and \({F_{{\gamma _{A}}}}(x) = {{\gamma ({m_{A}},x{\mu _{A}})} \over {\Gamma ({m_{A}})}}\) with \({\mu _{A}} = {m_{A}}/{\bar \gamma _{A}}\). Here, Γ(.) and γ(.,.) are the Gamma ([22] eq.(8.310.1)) and the lower incomplete Gamma ([22] eq.(8.350.1)) functions.
2.1 Fixedgain relaying
with \(C = {\bar \gamma _{SR}} + {\bar \gamma _{LI}} + 1\). Here, γ _{ LI }<C is obtained from the condition β ^{2}<1/(P _{ R }h _{ LI }^{2}) given before. Note that when the condition β ^{2}<1/(P _{ R }h _{ LI }^{2}) is not satisfied, i.e., when \({\gamma _{LI}} \geqslant C\), then γ _{ end }=0 since the power of the interference goes to infinity.
Although the instantaneous transmission power at the relay varies for the fixedgain AFFD relaying, the average value is set to P _{ R } due to the selected β. It is important to note that we assume the dynamic range of the power amplifier at the relay is unrestricted for the fixedgain AFFD systems similar to [8] and [21].
2.2 Variablegain relaying
which also shows that the average transmission power of the relay for both the fixedgain and variablegain AFFD relaying are the same and equal to P _{ R }.
3 Exact outage probability analysis
In this section, exact outage probabilities of the considered FD systems are analyzed by using the CDFs of γ _{ end } given in (5) and (8). The exact outage probability equals \(\phantom {\dot {i}\!}{P_{out}} = {F_{{\gamma _{end}}}}({\gamma _{th}})\) where \({F_{{\gamma _{end}}}}(.)\) is CDF of γ _{ end } and γ _{ th } is a threshold SINR.
3.1 Fixedgain relaying
The above expression is in a singleintegral form. Therefore, it can be easily and efficiently evaluated with common mathematical software packages such as MATLAB, MATHEMATICA or MAPLE.
3.2 Variablegain relaying
The above expression is in singleintegral form as well. Hence, it can be also easily and efficiently evaluated with common mathematical software packages such as MATLAB, MATHEMATICA, or MAPLE. As mentioned in the Introduction, outage performance of the variablegain AFFD relaying systems in Nakagamim fading channels is also independently evaluated in [20]. On the other hand, the exact outage probability expression given in ([20] eq. (16)) is written in terms of EGBMGF which contains MellinBarnes type doublecontourintegration ([24] Table I). Although, MATHEMATICA implementation of the EGBMGF function has been provided in ([24] Table II), the builtin function is not available for EGBMGF to the best of our knowledge. Also, different from [20], our result is valid for both integer and noninteger m _{ RD } values.
4 Lowerbound and asymptotic outage probability analyses
In this section, both the lowerbound and asymptotic outage probability of the fixedgain and variablegain AFFD systems are investigated.
4.1 Fixedgain relaying
4.1.1 Lowerbound
4.1.2 Asymptotic outage probability
As seen, \(P_{out}^{\infty } \) for the fixedgain case is a constant and therefore leads to zero diversity order at high SNR and does not depend on m _{ SR }, m _{ RD } and Ω _{ RD } parameters.
4.2 Variablegain relaying
4.2.1 Lowerbound
4.2.2 Asymptotic outage probability
As seen, \(P_{out}^{\infty } \) for the variablegain case is also a constant, hence leads to zero diversity order at high SNR. In addition, it does not depend on m _{ RD } and Ω _{ RD } parameters. Note that similar expression was also obtained independently in ([20] eq. (22)). Here, we provide (40) for the sake of consistency.
5 Numerical results
The theoretical (numerical) and MonteCarlo simulations outage performance results of the fixedgain and variablegain AFFD systems are shown in this section. During the outage performance analyses, we assume that \(\sigma _{D}^{2} = \sigma _{R}^{2}\) for all the scenarios and Ω _{ SR }=Ω _{ RD }=1, P _{ S }=P _{ R } unless otherwise stated. Moreover, the residual LI power, apart from Fig. 6, is chosen in the range of 0.01 and 0.1 (−20 and −10 dB) as in [12, 14, 18–20].
6 Conclusions
New exact outage probability expressions for the fixedgain and variablegain AFFD relaying systems were presented over nonidentical Nakagamim fading channels. The outage probabilities were obtained from CDF of the endtoend SINR at the destination and verified through MonteCarlo simulations. Our results show that the fixedgain AFFD relaying provides much better outage performance than the variablegain AFFD relaying when the power of the residual LI and SNR increase. We have also presented tight lowerbound and asymptotic outage performance results for both fixedgain and variablegain AFFD relaying. It was observed that the fixedgain AFFD relaying has much lower error floor with respect to the variablegain AFFD relaying. We have shown that m _{ LI } and m _{ SR } are dominant m parameters on the asymptotic outage performance of the fixedgain and variablegain AFFD relaying, respectively.
Declarations
Acknowledgements
The authors would like to thank the editor and anonymous reviewers for their constructive comments which helped to improve the technical quality of this paper.
Authors’ contributions
We, the authors of this paper, declare that all authors have materially participated in the research and the article preparation equally and have contributed equally to this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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