 Research
 Open Access
Performance analysis and power control for fullduplex relaying with largescale antenna array at the destination
 Liang Han^{1, 2}Email author,
 Bin Shi^{2} and
 Weixia Zou^{3}
https://doi.org/10.1186/s1363801708280
© The Author(s) 2017
 Received: 31 December 2016
 Accepted: 21 February 2017
 Published: 28 February 2017
Abstract
Fullduplex relaying can increase the spectral efficiency by allowing the relay to transmit and receive at the same time and over the same frequency. To overcome the challenge that the destination treats the sourcedestination link as interference, we employ largescale antenna array at the destination. By using the fact that the sourcedestination channel vector and relaydestination channel vector become pairwise orthogonal, the sourcedestination link can be converted to useful signal. We first derive the achievable rates and outage probabilities for direct transmission, halfduplex relaying, and fullduplex relaying, respectively. Two relaying protocols, i.e., amplifyandforward (AF) and decodeandforward (DF), are considered. Then, based on the channel state information (CSI) availability, the optimal transmit power of the fullduplex relay node is obtained by maximizing the achievable rates or minimizing the outage probabilities of fullduplex relaying. Numerical results show that the fullduplex relaying performs better than halfduplex relaying when the residual selfinterference is small, and the performance of fullduplex relaying can be improved by using power control.
Keywords
 Fullduplex relaying
 Amplifyandforward
 Decodeandforward
 Achievable rate
 Outage probability
 Power control
1 Introduction
In recent years, the fourthgeneration (4G) mobile communication system has been deployed and used worldwide. However, with an explosion of wireless devices and services, there is growing demand for data traffic that cannot be accommodated by 4G. Therefore, the fifthgeneration (5G) mobile communication system, which is expected to be commercially available by 2020, has become a hot research topic in both academia and industry [1, 2].
As one of the most promising technique for the 5G mobile communication system, largescale antenna array or massive multipleinput multipleoutput (MIMO) has drawn significant research interests [3–6]. The concept of massive MIMO was first proposed by Marzetta [7], showing that by employing unlimited number of antennas at a base station (BS), the effect of thermal noise and smallscale fading can be averaged out, the channel vectors between the users and the BS become pairwise orthogonal, and the interuser interference can be eliminated. In [8], the authors compared two linear precoding schemes, i.e., conjugate beamforming and zero forcing (ZF), with respect to spectral efficiency and energy efficiency in a singlecell downlink scenario, and showed that for high spectral efficiency and low energy efficiency, ZF outperforms conjugate beamforming, while at low spectral efficiency and high energy efficiency, the opposite holds. In [9], the transmit power scaling laws and lower capacity bounds for maximum ratio combining (MRC), ZF, and minimum meansquare error (MMSE) detection were derived. It was shown that when the number of BS antennas M grows without bound, the transmitted power of each user could be reduced proportionally to M if the BS has perfect channel state information (CSI) and 1/M if CSI is estimated from uplink pilots. In [10], the authors investigated how massive MIMO performs in real propagation environments and showed that the performance is close to that in independent and identically distributed (iid) Rayleigh channels, so the theoretical advantages of massive MIMO can be obtained in real channels.
On the other hand, encouraged by the advance in selfinterference cancelation (SIC) algorithms [11, 12], fullduplex relaying, which can transmit and receive at the same time and over the same frequency, has also attracted great research interests. In [13], the authors proposed a fullduplex relay system which shares the time resource as well as the antennas at the relay node and eliminates the underlying interferences using precoding. In [14], the exact outage probability of fullduplex decodeandforward (DF) relay systems was derived by considering the selfinterference as a Rayleigh fading channel. It was shown that the fullduplex relaying is superior to halfduplex relaying in terms of the outage probability as the signaltointerference ratios (SIRs) are higher and the signaltonoise ratio (SNR) is lower. Riihonen et al. derived an optimal gain control scheme for fullduplex amplifyandforward (AF) relaying which considered the effect of residual loop interference in [15], investigated three spatial domain SIC solutions for fullduplex MIMO relay systems in [16], and proposed a hybrid relaying which could switch between fullduplex and halfduplex mode in [17]. Reference [18] derived the outage performance of multihop fullduplex relaying by considering all the interrelay interference and echointerference. Reference [19] investigated the power and location optimization for fullduplex DF relay systems. From the above references, it is clear that there are two main challenges facing the use of fullduplex relaying. The first challenge is how to mitigate or cancel the selfinterference at the relay due to signal leakage between the relay transmit and receive antenna [12–19], and the second challenge is how to deal with the signal from the source since the destination always treats the direct link from the source to the destination as interference [14, 17–19].
Since low complexity linear signal processing would achieve near optimal performance in massive MIMO systems, some researchers have introduced massive antennas into fullduplex relay systems. In [20], the authors investigated a multipair fullduplex relay system, where the relay is equipped with massive arrays and uses ZF and MRC/MRT to process the signals, and showed that the large number of spatial dimensions available can be effectively used to suppress the loop interference in the spatial domain. In [21], the authors studied the hardware impairments aware transceiver design for massive MIMO fullduplex relaying, where multiple sourcedestination pairs communicate simultaneously with the help of a fullduplex relay equipped with very large antenna arrays. Note that these works assume the relay is equipped with massive arrays, and there is no work considering the destination is equipped with massive arrays in fullduplex relay systems.
In this paper, we investigate the performance of fullduplex AF and DF relaying with largescale antenna array at the destination. Since the sourcedestination channel vector and relaydestination channel vector become pairwise orthogonal, the destination can treat the sourcedestination link as useful signal and therefore the performance can be improved. Furthermore, we optimize the transmit power of the fullduplex relay node by maximizing the achievable rates or minimizing the outage probabilities.
2 System model
Denote the channel of the sourcerelay, sourcedestination, and relaydestination links as h _{sr}, \(\mathbf {h}_{{\text {sd}}}\in {\mathbb {C}^{N \times 1}}\), and \(\mathbf {h}_{{\text {rd}}}\in {\mathbb {C}^{N \times 1}}\), respectively, which are modeled to be frequency flat and quasi static. All the channel elements are assumed to be zeromean complex Gaussian random variables (Rayleigh fading), and the variance for h _{sr}, each element of h _{sd}, and each element of h _{rd} are given as η _{sr}, η _{sd}, and η _{rd}, respectively. Since the selfinterference cannot be completely canceled, we denote the residual selfinterference channel as h _{rr}, which is also assumed to be zeromean complex Gaussian random variables with variance η _{rr}.
The transmit signals from the source and the relay in time slot t are denoted as x _{s}(t) and x _{r}(t), respectively. We assume the transmit power of the source and the relay are P _{s} and P _{r}, respectively, i.e., E[x _{s}(t)^{2}]=P _{s} and E[x _{r}(t)^{2}]=P _{r}.
2.1 Direct transmission
where n _{d}(t) denotes the complex additive white Gaussian noise (AWGN) at the destination with zeromean and covariance matrix \(E\left [ {{{\mathbf {n}_{\mathrm {d}}}}\left (t \right){{\mathbf {n}_{\mathrm {d}}}}{{\left (t \right)}^{H}}} \right ] = \sigma _{\mathrm {d}}^{2}{\mathbf {I}_{N}}\).
2.2 Halfduplex relaying
When the DF protocol is used, the relay decodes its received signal and checks if it is correct. If the relay can decode the received signal successfully, the transmit signal at the relay is \({x_{\mathrm {r}}}\left ({t + 1} \right) = \sqrt {\frac {{{{P_{\mathrm {r}}}}}}{{{{P_{\mathrm {s}}}}}}} {x_{\mathrm {s}}}\left (t \right)\).
2.3 Fullduplex relaying
respectively.
Assume the processing delay at the relay is τ time slot, then the transmit signal at the relay is x _{r}(t)=β y _{r}(t−τ).
When the DF protocol is used, the processing at the relay is also similar to that in the halfduplex mode except there is processing delay τ. If the relay can decode the received signal successfully, then the transmit signal at the relay is \({x_{\mathrm {r}}}\left (t \right) = \sqrt {\frac {{{{P_{\mathrm {r}}}}}}{{{{P_{\mathrm {s}}}}}}} {x_{\mathrm {s}}}\left ({t  \tau } \right)\).
3 Performance analysis
In this section, we analyze the performance of the direct transmission, halfduplex relaying, and fullduplex relaying, respectively. We assume matched filter (MF) processing is used at the destination, i.e., the destination processes the signal vector by multiplying the conjugatetranspose of the channel [4]. For large N, the channel vectors h _{sd} and h _{rd} are asymptotically orthogonal, so we have hsdH h _{sd}→N η _{sd}, hrdH h _{rd}→N η _{rd}, hsdH h _{rd}→0 and hrdH h _{sd}→0.
3.1 Direct transmission
where \({{\bar \gamma }_{\mathrm {s,d}}} = {{P_{\mathrm {s}}}{\eta _{\mathrm {s,d}}}} \left / {\sigma _{\mathrm {d}}^{2}} \right.\).
3.2 Halfduplex relaying
where \({{\bar \gamma }_{{\text {sr}}}} = {{P_{\mathrm {s}}}{\eta _{{\text {sr}}}}} \left / {\sigma _{\mathrm {r}}^{2}} \right.\).
3.3 Fullduplex relaying
where \({{\bar \gamma }_{{\text {rr}}}} = {{P_{\mathrm {r}}}{\eta _{{\text {rr}}}}} \left / {\sigma _{\mathrm {r}}^{2}} \right.\).
3.4 Condition for the superiority of fullduplex relaying over halfduplex relaying

Case 1: \(N{{\bar \gamma }_{{\text {sd}}}} \ge {\delta _{{\text {HD}}}}\). In this case, we have pHD,AFout=pHD,DFout=pFD,AFout=pFD,DFout=0, which means that fullduplex relaying and halfduplex relaying will obtain the same outage performance.

Case 2: \(N{{\bar \gamma }_{{\text {sd}}}} + N{{\bar \gamma }_{{\text {rd}}}} < {\delta _{{\text {FD}}}}\). In this case, we have pHD,AFout=pHD,DFout=pFD,AFout=pFD,DFout=1, which means that fullduplex relaying and halfduplex relaying will obtain the same outage performance.

Case 3: \({\delta _{{\text {FD}}}} \le N{{\bar \gamma }_{{\text {sd}}}} < {\delta _{{\text {HD}}}}\). In this case, we have pFD,AFout=pFD,DFout=0 and pHD,AFout=pHD,DFout>0, so fullduplex relaying outperforms halfduplex relaying.

Case 4: \({\delta _{{\text {FD}}}} \le N{{\bar \gamma }_{{\text {sd}}}} + N{{\bar \gamma }_{{\text {rd}}}} < {\delta _{{\text {HD}}}}\). In this case, we have pHD,AFout=pHD,DFout=1 and pFD,AFout=pFD,DFout<1, so fullduplex relaying outperforms halfduplex relaying.

Case 5: \(N{{\bar \gamma }_{{\text {sd}}}} + N{{\bar \gamma }_{{\text {rd}}}} \ge {\delta _{{\text {HD}}}}\) and \(N{{\bar \gamma }_{{\text {sd}}}} < {\delta _{{\text {FD}}}}\). In this case, fullduplex relaying outperforms halfduplex relaying when pFD,AFout≤pHD,AFout for AF protocol and pFD,DFout≤pHD,DFout for DF protocol. Using the results (15) and (27), the average residual selfinterference for AF protocol should satisfy$$\begin{array}{*{20}l} {{\bar \gamma }_{{\text{rr}}}} \le & \frac{{{{\bar \gamma }_{{\text{sr}}}}\left({N{{\bar \gamma }_{{\text{sd}}}} + N{{\bar \gamma }_{{\text{rd}}}}  {\delta_{{\text{FD}}}}} \right)}}{{\left({N{{\bar \gamma }_{{\text{rd}}}} + 1} \right)\left({{\delta_{{\text{FD}}}}  N{{\bar \gamma }_{{\text{sd}}}}} \right)}}\\ & \times \left\{ {\exp \left[ {\frac{{N{{\bar \gamma }_{{\text{rd}}}}\left({N{{\bar \gamma }_{{\text{rd}}}} + 1} \right)}}{{{{\bar \gamma }_{{\text{sr}}}}\left({N{{\bar \gamma }_{{\text{sd}}}} + N{{\bar \gamma }_{{\text{rd}}}}  {\delta_{{\text{HD}}}}} \right)}}} \right.} \right.\\ &\left. { \times \left. {\frac{{\left({{\delta_{{\text{HD}}}}  {\delta_{{\text{FD}}}}} \right)}}{{\left({N{{\bar \gamma }_{{\text{sd}}}} + N{{\bar \gamma }_{{\text{rd}}}}  {\delta_{{\text{FD}}}}} \right)}}} \right]  1} \right\}. \end{array} $$(32)Using the results (19) and (31), the average residual selfinterference for DF protocol should satisfy$$ {{\bar \gamma }_{{\text{rr}}}} \le \frac{{{{\bar \gamma }_{{\text{sr}}}}}}{{{\delta_{{\text{FD}}}}}}\left[ {\exp \left({\frac{{{\delta_{{\text{HD}}}}  {\delta_{{\text{FD}}}}}}{{{{\bar \gamma }_{{\text{sr}}}}}}} \right)  1} \right]. $$(33)
4 Power control
In this section, we propose power control to improve the system performance. We assume independent power constraint at both the source and the relay, i.e., \({P_{\mathrm {s}}} \le P_{\mathrm {s}}^{\max }\) and \({P_{\mathrm {r}}} \le P_{\mathrm {r}}^{\max }\). For halfduplex relaying, it is obvious that the system performance is optimized when the source and the relay use the maximum allowed transmit power, so we focus on power control for fullduplex relaying.
For fullduplex relaying, the source should also use the maximum transmit power because as P _{s} increases, the achievable rates (25) and (29) increase and the outage probabilities (27) and (31) decrease. As such, we concentrate on power control for the relay. Two scenarios are considered based on the CSI availability.
4.1 Instantaneous CSI of h _{ sr } and h _{ rr }
In this scenario, we assume the relay has the instantaneous CSI of h _{sr} and h _{rr}, and the statistical CSI of h _{sd} and h _{rd}, then the power control is to maximize the achievable rates.
Therefore, the optimal transmit power of the relay is \(P_{\mathrm {r}}^{\text {opt}} = \min \left ({P_{\mathrm {r}}^{*},P_{\mathrm {r}}^{\max }} \right)\).
Similarly, the optimal transmit power of the relay for DF protocol is \(P_{\mathrm {r}}^{\text {opt}} = \min \left ({P_{\mathrm {r}}^{*},P_{\mathrm {r}}^{\max }} \right)\).
4.2 Statistical CSI of h _{sr} and h _{ rr }
In this scenario, we assume the relay only has the statistical CSI of h _{sr}, h _{rr}, h _{sd,} and h _{rd}, then the power control is to minimize the outage probabilities. Note that power control does not help if \(N{{\bar \gamma }_{{\text {sd}}}} \ge {\delta _{{\text {FD}}}}\) or \(N{{\bar \gamma }_{{\text {sd}}}} + N{{\bar \gamma }_{{\text {rd}}}} < {\delta _{{\text {FD}}}}\), so we only need to consider the case that \(N{{\bar \gamma }_{{\text {sd}}}} < {\delta _{{\text {FD}}}}\) and \(N{{\bar \gamma }_{{\text {sd}}}} + N{{\bar \gamma }_{{\text {rd}}}} \ge {\delta _{{\text {FD}}}}\).
where \(c = {N \left / {\left ({{\delta _{{\text {FD}}}}  N{{\bar \gamma }_{{\text {sd}}}}} \right)}\right.} \). By taking the first derivative with respect to P _{r} and setting the first derivative equal to zero, we can obtain \(P_{\mathrm {r}}^{*}\). However, it is difficult to find the closed form solution, so we need to use iterative rootfinding algorithms such as the bisection method or the Newton’s method to find the numerical solution.
5 Numerical results
In this section, we present some numerical results to validate our analysis. For simplicity, we assume N=100, \(P_{\mathrm {s}}^{{\text {max}}} = 1\), \(P_{\mathrm {r}}^{{\text {max}}} = 1\), η _{sd}=10^{−7}, η _{rd}=2×10^{−6}, \(\sigma _{\mathrm {d}}^{2} = \sigma _{\mathrm {r}}^{2} = {10^{ 6}}\), and R _{0}=5 bps/Hz.
6 Conclusions
In traditional fullduplex relaying, the destination always treats the sourcedestination link as interference, which degrades the system performance. In this paper, we investigated the system performance of fullduplex relaying by considering largescale antenna array at the destination. Using the property that the sourcedestination channel vector and relaydestination channel vector become pairwise orthogonal, we converted the sourcedestination link to useful signal. We derived the achievable rates and outage probabilities for direct transmission, halfduplex relaying, and fullduplex relaying, respectively. By maximizing the achievable rates or minimizing the outage probabilities based on the CSI availability, we obtained the optimal transmit power of the fullduplex relay node. Numerical results showed that the performance of fullduplex relaying is determined by the residual selfinterference, and power control can improve the performance of fullduplex relaying.
Declarations
Acknowledgements
This work was supported by the 863 Program of China (2015AA01A703), the Fundamental Research Funds for the Central Universities (2014ZD0302), the National Natural Science Foundation of China (61571055), the China Scholarship Council (201508120012), and the Doctoral Foundation of Tianjin Normal University (5RL135).
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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