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Performance analysis of two-way full-duplex relay with antenna selection under Nakagami channels


This paper focuses on a full-duplex (FD) two-way relay network with one base station (BS), one FD amplify-and-forward (AF) relay, and one user, and the BS is equipped with massive multiple-input multiple-output (MIMO) antennas. To reduce the complexity and cost of BS, the paper proposes a practical antenna selection way in BS to complete transmission in two-way FD relay networks to optimize the system performance of outage probability or bit error ratio (BER). In the proposed scheme, BS can implement transmit antenna selection (AS) based on the instantaneous channel state information (CSI), thereby improving the system performance compared to the traditional FD relay networks. Furthermore, closed-form expressions for the outage probability and average BER are derived under Nakagami-m channels. The analytical results show that the superiority of proposed scheme compared with the conventional AF relay scheme and the correctness of the theoretical analysis verified by the Monte Carlo simulations.

1 Introduction

MIMO has been adopted by various wireless standards, such as Third Generation Partnership Project (3GPP), long-term evolution (LTE), and 5G communications, as a promising way to boost system throughput and improve coverage range. Massive MIMO systems employing simple linear precoding and combining schemes can offer significant performance gains in terms of bandwidth, power, and energy efficiency compared to conventional multiuser MIMO systems, as impairments such as fading, noise, and interference are averaged out for very large numbers of base station (BS) antennas. Antenna diversity is one of the most effective techniques to combat the multipath fading and shadowing in wireless communications [1,2,3]. However, each additional antenna comes along with an additional radio frequency (RF) chain and power consumption, which may increase the transmission cost and complexity of the system as a whole. A remedy for this crucial issue is the use of antenna selection, which can reduce system cost and complexity and improve the system error performance considerably through selecting a single antenna or a group of antennas [4,5,6,7,8].

Relay technologies are also included in the current standard for wireless metropolitan area networks such as IEEE 802.16j and IEEE 802.16 m and are adopted by cellular communication systems such as 3GPP and LTE-Advanced [9, 10]. There has been lots of research on antenna selection in AF relay networks recently [11,12,13,14,15]. In another study, Zheng provided a semi-analytical result on the error performance of the fractionally spaced frequency domain and minimum mean square error receiver over flat Nakagami-m fading channels and derived the bit error rate (BER) and outage probability [16]. You and co-authors proposed dual antenna selection strategies for MIMO scenarios with partial feedback based on system outage probability minimization and sum rate maximization of two-way MIMO AF relay networks [17,18,19]. Some authors also investigated the beamforming schemes and interference alignment in cognitive networks [20, 21].

In more recent years, one-way DF relay considered with AS has also garnered a great deal of research interest. Lv and co-author investigated the max-min method for relay AS in two-way DF relay systems and proposed a single antenna system with receiver optimization [22, 23]. Their approach can be implemented in a distributed manner without acquiring full channel state information (CSI) at the relay node. Meanwhile, some authors also investigate other key technologies about 5G except the full-duplex system [24,25,26,27].

For conventional bidirectional communications, most researchers to date have focused primarily on half-duplex (HD) relay. Recently, with the advent of multi-antenna technologies, there has been a renewed interest in the use of the FD relay [28, 29]. In one such study, Krikidis proposed an optimal relay selection procedure that incorporates a hybrid relaying strategy which dynamically switches between FD and HD relaying according to the instantaneous channel conditions [28]; this system well outperformed the conventional FD relay system. In another study, Riihonen attempted to maximize the D2D information rate from the source to the destination for the FD-AF relay with multiple antennas of transmitter and receiver [29]. The optimal transformation matrix of the relay was derived under the assumption that the self-interference is perfectly canceled by time domain digital signal processing.

The remainder of the paper is organized as follows. First, in Section 2, we elaborate the research methods and contributions in the paper. Section 3 describes the system model. Then, we propose the max-min antenna selection scheme in Section 4. In Section 4, we derive the performance of antenna selection for two-way FD-AF relay networks in the flat Nakagami-m fading channel. Simulation results are given in Section 5, and Section 6 concludes the paper.

2 Methods/experimental

There has been very extensive research on FD relay in 5G networks. Merra et al. analyzed the multiband cognitive radio full-duplex relay and obtained the outage probability expressions [30]. Zhang et al. explored the performance of light communication networks with full-duplex optical links and proposed two contention protocols to utilize the FD capability effectively [31]. Atzeni used stochastic geometry to analyze the performance of wireless networks with FD multi-antenna small cells and emphasis on the probability of successful transmission [32]. It is important to note that all of these studies explored transmission protocol and performance through stochastic geometry; performance analyses on the two-way FD-AF relay with antenna selection in 5G systems are, at present, few and far between. This paper emphasizes this perspective from which we attempted to achieve outage probability and BER expressions.

The full-duplex system can achieve two times the spectrum gains of the half duplex. However, how does the performance of the FD relay system fare in terms of antenna selection? To the best of our knowledge, performance analyses of two-way FD-AF relay networks in terms of antenna selection are scarce in recent studies. The performance analysis of massive MIMO combined with antenna selection in the FD relay network was derived in this study with various channel models. In our previous work, we have investigate the performance of antenna selection with AF and DF half-duplex relay networks [3], our main contributions in this paper can be summarized as follows.

  1. 1)

    We propose an antenna selection scheme for FD-AF relay networks based on the max-min criterion.

  2. 2)

    We derived the closed-form outage probability and BER expressions in flat Nakagami-m fading channels. Our analysis shows that the proposed scheme achieves superior performance compared to other conventional schemes.

3 System model

We consider two-way FD-AF relay networks as shown in Fig. 1, where S1, R, and S2 are equipped with N1, NR, and N2 antennas, respectively. H and G are channel matrix of links S1R and S2R, respectively. We assume all elements of H and G are independent and identically distributed (i.i.d.) flat Nakagami-m fading. It is worth noting that the Nakagami-m fading is a general case, which can cover Rayleigh fading or Rician fading and so on, such as m = 1 means Nakagami-m is equal to Rayleigh fading. There is no direct link between S1 and S2, and all nodes are working in a full-duplex mode. Based on known CSI, the BS select one of their antennas separately for transmission. The work process of the two-way FD-AF relay networks can be seen as follows.

Fig. 1
figure 1

The system model

The relay always amplifies and forwards the received signal in every time slot because of its full-duplex mode. Therefore, we denote n as time slot number. In nth time slot, the BS S1 and user S2 send their information x1[n] and x2[n] simultaneously, then the relay node R amplifies and forwards the received signal in the (n + 1)th time slot. In time slot n, BS selects the ith antenna through the max-min criterion in the following, then the received signal in relay R can be expressed as follows:

$$ {y}_R\left[n\right]=\sqrt{P_1}{h}_i{x}_1\left[n\right]+\sqrt{P_2}{gx}_2\left[n\right]+\sqrt{P^{(R)}}{h}^{(R)}t\left[n\right]+{n}_R\left[n\right] $$

where hi is the ith column of h, t[n] is the forward signal, and nR is additive Gaussian white noise of R and satisfies \( {n}_R\sim \mathcal{CN}\left(0,{\sigma}_R^2\right) \), where \( \mathcal{CN}\left(\mu, {\sigma}^2\right) \) represents circular symmetric complex Gaussian distribution of μ mean and σ2 variance.

For the full-duplex relay network, the forward signal t[n] is determined by the received signal yR[n − 1]. Due to the power constraint of the relay node, the received signal yR then needs to multiply factor α as follows:

$$ \alpha =\sqrt{\frac{1}{P_1{\left\Vert {h}_i\right\Vert}^2+{P}_2{\left\Vert g\right\Vert}^2+{P}^{(R)}{\left\Vert {h}^{(R)}\right\Vert}^2+{\sigma}_R^2}} $$

In the nth time slot, R sends the amplified signal αyR[n].

Based on the channel reciprocity in the TDD (time division duplex) system, the signal in nth time slot can be rewritten as follows:

$$ {\displaystyle \begin{array}{c}{y}_1\left[n\right]=\sqrt{P_R}\alpha {h}_i^{\ast }{y}_R\left[n-1\right]+\sqrt{P^{(1)}}{h}_i^{(1)}{x}_1\left[n\right]+{n}_1\left[n\right]\\ {}{y}_2\left[n\right]=\sqrt{P_R}\alpha {g}^{\ast }{y}_R\left[n-1\right]+\sqrt{P^{(2)}}{h}_i^{(2)}{x}_2\left[n\right]+{n}_2\left[n\right]\end{array}} $$

where () stands for the conjugate transpose and n1 and n2 represent the additive white Gaussian noise for B and U, respectively, and satisfy \( {n}_1\sim \mathcal{CN}\left(0,{\sigma}_1^2\right) \) and \( {n}_2\sim \mathcal{CN}\left(0,{\sigma}_2^2\right) \).

After self-cancelation for B and U, the equivalent signal can be expressed as follows:

$$ {\displaystyle \begin{array}{c}\begin{array}{l}{\tilde{y}}_1\left[n\right]=\sqrt{P_R{P}_2}\alpha {h}_i^{\ast }{gx}_2\left[n\right]+\sqrt{P_R{P}^{(R)}}\alpha {h}_i^{\ast }{h}^{(R)}t\left[n\right]\\ {}+\sqrt{P_R}\alpha {h}_i^{\ast }{n}_R\left[n\right]+\sqrt{P^{(1)}}{h}_i^{(1)}{x}_1\left[n\right]+{n}_1\left[n\right]\end{array}\\ {}\begin{array}{l}{\tilde{y}}_2\left[n\right]=\sqrt{P_R{P}_1}\alpha {g}^{\ast }{h}_i{x}_1\left[n\right]+\sqrt{P_R{P}^{(R)}}\alpha {g}^{\ast }{h}^{(R)}t\left[n\right]\\ {}+\sqrt{P_R}\alpha {g}^{\ast }{n}_R\left[n\right]+\sqrt{P^{(2)}}{h}^{(2)}{x}_2\left[n\right]+{n}_2\left[n\right]\end{array}\end{array}} $$

Without loss of generality, we assume \( {\sigma}_R^2={\sigma}_1^2={\sigma}_2^2=1 \) and the self-interference channel of each node varies slowly relative to the transceiver channel. The SNR of B and U can be obtained as follows:

$$ {\displaystyle \begin{array}{l}{\gamma}_1=\frac{{\tilde{P}}_2{\tilde{P}}_R^{(1)}{\left\Vert {h}_i\right\Vert}^2{\left\Vert g\right\Vert}^2}{\left({\tilde{P}}_R^{(1)}+{\tilde{P}}_1\right){\left\Vert {h}_i\right\Vert}^2+{\tilde{P}}_2{\left\Vert g\right\Vert}^2+1}\\ {}{\gamma}_2=\frac{{\tilde{P}}_1{\tilde{P}}_R^{(2)}{\left\Vert {h}_i\right\Vert}^2{\left\Vert g\right\Vert}^2}{{\tilde{P}}_1{\left\Vert {h}_i\right\Vert}^2+\left({\tilde{P}}_R^{(2)}+{\tilde{P}}_2\right){\left\Vert g\right\Vert}^2+1}\end{array}} $$


$$ {\tilde{P}}_1=\frac{P_1}{P^{(R)}{\left({\sigma}^{(R)}\right)}^2+1} $$
$$ {\tilde{P}}_2=\frac{P_2}{P^{(R)}{\left({\sigma}^{(R)}\right)}^2+1} $$
$$ {\tilde{P}}_R^{(1)}=\frac{P_R}{P^{(1)}{\left({\sigma}_i^{(1)}\right)}^2+1} $$
$$ {\tilde{P}}_R^{(2)}=\frac{P_R}{P^{(2)}{\left({\sigma}_k^{(2)}\right)}^2+1} $$

The corresponding channel variance can be defined as \( {\left({\sigma}_j^{(R)}\right)}^2=E\left\{{\left\Vert {h}^{(R)}\right\Vert}^2\right\} \), \( {\left({\sigma}_i^{(1)}\right)}^2=E\left\{{\left\Vert {h}_i^{(1)}\right\Vert}^2\right\} \), and \( {\left({\sigma}_k^{(2)}\right)}^2=E\left\{{\left\Vert {h}^{(2)}\right\Vert}^2\right\} \), where E{} represents random variable expectation operation.

The BS works under an optimal antenna selection scheme where the optimal antenna as selected can be used to transmit and receive signals. The selection criterion can be expressed as follows:

$$ \overset{\frown }{i}=\arg \underset{i}{\max}\min \left\{{\gamma}_1,{\gamma}_2\right\} $$

The selection criterion can be modified as follows via some mathematical manipulation:

$$ \overset{\frown }{i}=\arg \underset{i}{\max }{\left\Vert {h}_i\right\Vert}^2 $$

It is worth noting that the value of hi2may be very large when massive MIMO BS includes optimal antenna selection. Therefore, γ1and γ2 can be approximated as:

$$ {\displaystyle \begin{array}{l}{\gamma}_1\approx \frac{{\tilde{P}}_2{\tilde{P}}_R^{(1)}{\left\Vert {h}_i\right\Vert}^2{\left\Vert g\right\Vert}^2}{\left({\tilde{P}}_R^{(1)}+{P}_1\right){\left\Vert {h}_i\right\Vert}^2+{\tilde{P}}_2{\left\Vert g\right\Vert}^2}\\ {}{\gamma}_2\approx \frac{{\tilde{P}}_1{\tilde{P}}_R^{(2)}{\left\Vert {h}_i\right\Vert}^2{\left\Vert g\right\Vert}^2}{{\tilde{P}}_1{\left\Vert {h}_i\right\Vert}^2+\left({\tilde{P}}_R^{(2)}+{\tilde{P}}_2\right){\left\Vert g\right\Vert}^2}\end{array}} $$

4 Performance analysis

4.1 PDF of end-to-end SNR

In this section, we derive probability density function (PDF) in high SNR approximation based on end-to-end SNR analysis and outage probability. We assume that \( {\left\Vert {h}_i\right\Vert}^2\sim \mathcal{CN}\left(0,{\sigma}_h^2\right) \) and g2~Gamma(α, β), then the PDF of hi2and g2can be expressed as [33]:

$$ {f}_{{\left\Vert {h}_i\right\Vert}^2}(x)=\frac{1}{\sigma_h^2}{e}^{-\frac{x}{\sigma_h^2}} $$
$$ {f}_{{\left\Vert g\right\Vert}^2}(y)=\frac{\beta^{\alpha }}{\Gamma \left(\alpha \right)}{y}^{\alpha -1}{e}^{-\beta y} $$

It is worth noting that channel g follows Nakagami channel, and its norm square obeys the Gamma distribution. Nakagami-m channel can be equivalent to other wireless multipath fading channels with different parameters \( m\in \left(\frac{1}{2},\infty \right) \), which present that Nakagami-m channel can cover other fading channels generally. In other words, Nakagami channel model has become a general channel model and thus has a high application value such as

$$ \mathrm{Nakagami}\hbox{-} m\ \mathrm{channel}=\left\{\begin{array}{c}\ \mathrm{Unilateral}\ \mathrm{Gaussian}\ \mathrm{channel},\kern3.75em m=\frac{1}{2}\\ {}\mathrm{Rayleigh}\ \mathrm{channel},\kern7.75em m=1\\ {}\mathrm{Nagami}\hbox{-} n,m=\frac{{\left(1+{n}^2\right)}^2}{1+2{n}^2},\kern5.25em n\ge 0\\ {}\mathrm{Rice}\ \mathrm{channel},K={n}^2=\frac{\sqrt{m^2-m}}{m-\sqrt{m^2-m}},m\ge 1\\ {}\mathrm{Constant}\ \mathrm{channel},\kern8em m=\infty \end{array}\right.. $$

The BS can achieve optimal antenna selection after finishing the antenna selection scheme. The PDF and CDF of equivalent channel h2 = maxihi2can then be expressed as:

$$ {f}_{{\left\Vert h\right\Vert}^2}(x)=\frac{N_B}{\sigma_h^2}{\left(1-{e}^{-\frac{x}{\sigma_h^2}}\right)}^{N_B-1}{e}^{-\frac{x}{\sigma_h^2}} $$
$$ {F}_{{\left\Vert h\right\Vert}^2}(x)={\left(1-{e}^{-\frac{x}{\sigma_h^2}}\right)}^{N_B} $$

Because channels h and g are independent, then the joint probability density function of h and g can be calculated as follows:

$$ {p}_{{\left\Vert h\right\Vert}^2,{\left\Vert g\right\Vert}^2}\left(x,y\right)=\frac{N_B}{\sigma_h^2}\cdot \frac{\beta^{\alpha }}{\Gamma \left(\alpha \right)}{\left(1-{e}^{-\frac{x}{\sigma_h^2}}\right)}^{N_B-1}\cdot {e}^{-\frac{x}{\sigma_h^2}}\cdot {y}^{\alpha -1}\cdot {e}^{-\beta y} $$

We can define two new variables to derive the PDF of γ1 and γ2.

We can define \( z=\frac{xy}{ax+ by} \), \( \omega =\frac{ax^2}{ax+ by} \). The Jacobian determinant can then be expressed as follows [34]:

$$ \det \left({J}_F\left(z,\omega \right)\right)=\det \left[\begin{array}{cc}\frac{\partial x}{\partial z}& \frac{\partial x}{\partial \omega}\\ {}\frac{\partial y}{\partial z}& \frac{\partial y}{\partial \omega}\end{array}\right]=a+\frac{2 abz}{\omega }+\frac{ab^2{z}^2}{\omega^2} $$

where the joint PDF of Z and W is:

$$ {\displaystyle \begin{array}{l}{p}_{Z,W}\left(z,\omega \right)=\det \left({J}_F\left(z,\omega \right)\right)\frac{N_B}{\sigma_h^2}\cdot \frac{\beta^{\alpha }}{\Gamma \left(\alpha \right)}\cdot \\ {}\sum \limits_{j=0}^{N_B-1}{C}_{N_B-1}^j{\left(-1\right)}^j{\left[ az+\frac{abz^2}{\omega}\right]}^{\alpha -1}\cdot {e}^{-\frac{j+1}{\sigma_h^2}\left(\omega + bz\right)}\cdot {e}^{-\beta \left[ az+\frac{abz^2}{\omega}\right]}\end{array}} $$

then we can derive the PDF of Z as:

$$ {p}_Z(z)={\int}_0^{\infty }{p}_{ZW}\left(z,\omega \right) d\omega $$

Substitute the joint PDF pZW(z, ω) to the integral of pZ(z):

$$ {\displaystyle \begin{array}{l}{p}_Z(z)={\int}_0^{\infty}\frac{N_B}{\sigma_h^2}\cdot \frac{\beta^{\alpha }}{\Gamma \left(\alpha \right)}\cdot \sum \limits_{j=0}^{N_B-1}{\left(-1\right)}^j\cdot \left[a+\frac{2 abz}{\omega }+\frac{ab^2{z}^2}{\omega^2}\right]\cdot \\ {}{\left[ az+\frac{{ab z}^2}{\omega}\right]}^{\alpha -1}\cdot {e}^{-\frac{j+1}{\sigma_h^2}\cdot \left(\omega + bz\right)}\cdot {e}^{-\beta \left[ az+\frac{{ab z}^2}{\omega}\right]} d\omega \end{array}} $$

Then, the integral of pZ(z) can be divided into three parts:

$$ {p}_Z(z)=\frac{N_B}{\sigma_h^2}\cdot \frac{\beta^{\alpha }}{\Gamma \left(\alpha \right)}\cdot {\int}_0^{\infty}\left({I}_1+{I}_2+{I}_3\right) d\omega $$

where I1, I2, and I3 can be represented as follows:

$$ {I}_1=a\cdot {\left[ az+\frac{abz^2}{\omega}\right]}^{\alpha -1}\cdot {e}^{-\frac{j+1}{\sigma_h^2}\cdot \left(\omega + bz\right)}\cdot {e}^{-\beta \left[ az+\frac{abz^2}{\omega}\right]} $$
$$ {I}_2=\frac{2 abz}{\omega}\cdot {\left[ az+\frac{abz^2}{\omega}\right]}^{\alpha -1}\cdot {e}^{-\frac{j+1}{\sigma_h^2}\cdot \left(\omega + bz\right)}\cdot {e}^{-\beta \left[ az+\frac{abz^2}{\omega}\right]} $$
$$ {I}_3=\frac{ab^2{z}^2}{\omega^2}\cdot {\left[ az+\frac{{ab z}^2}{\omega}\right]}^{\alpha -1}\cdot {e}^{-\frac{j+1}{\sigma_h^2}\cdot \left(\omega + bz\right)}\cdot {e}^{-\beta \left[ az+\frac{{ab z}^2}{\omega}\right]} $$

Therefore, we can determine the PDF of \( z=\frac{xy}{ax+ by} \) as:

$$ {\displaystyle \begin{array}{l}{p}_Z(z)=\frac{N_B}{\sigma_h^2}\cdot \frac{\beta^{\alpha }}{\Gamma \left(\alpha \right)}\cdot \sum \limits_{j=0}^{N_B-1}\sum \limits_{k=0}^{\alpha -1}{C}_{N_B-1}^j{C}_{\alpha -1}^k{\left(-1\right)}^j\cdot {\left(\frac{j+1}{\sigma_h^2}\right)}^{\frac{k}{2}}\cdot {e}^{-\left[\frac{j+1}{\sigma_h^2}b+\beta a\right]z}\cdot {\beta}^{\frac{-k}{2}}\cdot {a}^{\alpha -\frac{k}{2}}\cdot {b}^{\frac{k}{2}}\cdot {z}^{\alpha}\cdot \\ {}\left\{{\left(\frac{j+1}{\sigma_h^2}\cdot \frac{b}{\beta a}\right)}^{-\frac{1}{2}}2{z}^{-1}{K}_{k-1}\left(2z\sqrt{\beta a b\frac{j+1}{\sigma_h^2}}\right)+4{bK}_k\left(2z\sqrt{\beta a b\frac{j+1}{\sigma_h^2}}\right)+\right.\\ {}\left.{\left(\frac{j+1}{\sigma_h^2}\cdot \frac{1}{\beta a}\right)}^{\frac{1}{2}}{b}^{\frac{3}{2}}2{z}^{-2}{K}_{k+1}\left(2z\sqrt{\beta a b\frac{j+1}{\sigma_h^2}}\right)\right\}\end{array}} $$

where the special function Kα(x) of PDF is modified Bessel function, the Bessel functions are valid even for complex arguments x, and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions of the second kind. It is worth noting that the parameter α in the special function Kα(x) is not an integer, when α is an integer, then the limit is used, these are chosen to be real-valued for real and positive arguments x.

4.2 Outage probability

As established by the researchers referenced above, any link SNR lower than the threshold can result in system outage. The outage probability of the system model can be expressed as:

$$ {\displaystyle \begin{array}{l}{P}_{\mathrm{out}}=\Pr \left\{\min \left({\gamma}_1,{\gamma}_2\right)<{\gamma}_{\mathrm{th}}\right\}\\ {}=1-\left(1-{P}_{\gamma_1}\left({\gamma}_{\mathrm{th}}\right)\right)\left(1-{P}_{\gamma_2}\left({\gamma}_{\mathrm{th}}\right)\right)\end{array}} $$

where \( {P}_{\gamma_1}(x) \) and \( {P}_{\gamma_2}(x) \) represent the cumulative distribution function (CDF) of two end nodes’ SNR.

The CDF of \( {P}_{\gamma_1}(x) \) and \( {P}_{\gamma_2}(x) \) can be obtained based on the derived PDF; however, the expressions of \( {P}_{\gamma_1}(x) \) and \( {P}_{\gamma_2}(x) \) are complex. We can achieve the approximation expression of Poutusing the relationship between system outage probability and parameters:

$$ {P}_{\mathrm{out}}=1-\left[1-{\left(1-{e}^{-\frac{\gamma_{\mathrm{th}}}{{\tilde{\Omega}}_1}}\right)}^{N_B}\right]\cdot \frac{\Gamma \left(\alpha, {\beta \gamma}_{\mathrm{th}}\right)}{\Gamma \left(\alpha \right)} $$

where \( {\tilde{\Omega}}_1=\frac{{\tilde{P}}_R{\tilde{P}}_2}{{\tilde{P}}_R+{\tilde{P}}_1}{\theta}_1 \).

The BS antenna number is greater than users’ in massive MIMO system, so we can obtain the following corollary:

Corollary 1 When the BS antenna number is larger than users’ (NB >  > 1), then the outage probability can be approximated as follows:

$$ {P}_{\mathrm{out}}\approx 1-\frac{\Gamma \left(\alpha, {\beta \gamma}_{\mathrm{th}}\right)}{\Gamma \left(\alpha \right)} $$

It can be elaborated the proof process in a simple mode, when NB >  > 1, then \( 1-{e}^{-\frac{\gamma_{\mathrm{th}}}{{\tilde{\Omega}}_1}}<1 \), we can obtain

$$ \underset{N_B\to \infty }{\lim }{\left(1-{e}^{-\frac{\gamma_{\mathrm{th}}}{{\tilde{\Omega}}_1}}\right)}^{N_B}=0 $$

Then, the corollary can be achieved normally.

It is worth noting that the outage probability of a massive MIMO full-duplex system is limited to the link between relay and user, the reason being that the link between the BS and relay tends to be idealized when the BS antenna number grows to infinity. The link quality between relay and user shows unobvious improvement, however.

4.3 Average BER

It is known that the average BER can be defined as the average level of each link BER. We assume that P1 is the BER of link B → R → U and P2 is the BER of link U → R → B. The average BER is decided by the worst link BER. We can then rewrite Pe as follows:

$$ {P}_e\approx \frac{1}{2}\max \left({P}_1,{P}_2\right) $$

Theorem 2 The average BER of the optimal antenna selection scheme can be expressed as:

$$ {\displaystyle \begin{array}{l}{P}_e\approx \frac{1}{2}\max \left({P}_1,{P}_2\right)\\ {}=\frac{t_1\sqrt{t_2}}{4\sqrt{\pi }}\frac{\beta^{\alpha}\Gamma \left(\alpha +\frac{1}{2}\right)}{{\left(\beta +{t}_2\right)}^{\alpha +\frac{1}{2}}\Gamma \left(\alpha +1\right)}{{}_2F}_1\left(1,\alpha +\frac{1}{2};\alpha +1;\frac{\beta }{\beta +{t}_2}\right)+\\ {}\frac{t_1\sqrt{t_2}}{4\sqrt{\pi }}\sum \limits_{i=0}^{N_B}{C}_{N_B}^i{\left(-1\right)}^i\frac{2{\beta}^{\alpha}\Gamma \left(\alpha +\frac{1}{2}\right)}{{\left(\beta +{t}_2+\frac{i}{{\tilde{\Omega}}_1}\right)}^{\alpha +\frac{1}{2}}\Gamma \left(\alpha \right)}{{}_2F}_1\left(1,\alpha +\frac{1}{2};\frac{3}{2};\frac{t_2+\frac{i}{{\tilde{\Omega}}_1}}{\beta +{t}_2+\frac{i}{{\tilde{\Omega}}_1}}\right)\end{array}} $$

The proof can be seen as Appendix. Where the special function 2F1(a, b; c; z) in the Pe is the hypergeometric function, the Gaussian or ordinary hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation and can be defined as \( {}_2{F}_1\left(a,b;c;z\right)=\sum \limits_{n=0}^{\infty}\frac{(a)_n{(b)}_n}{(c)_n}\frac{z^n}{n!} \).

Therefore, this section derived the full-duplex two-way relay system end node’s SNR PDF. On the basis of this derivation, we further investigated the system outage probability and asymptotic BER expressions as discussed below.

5 Results and discussion

This section focuses on verifying the effectiveness of the proposed scheme and the correctness of said performance analysis as well as the discussion of the results.

The outage probability and BER of the two-way FD-AF relay network with antenna selection were simulated and analyzed in detail. The simulation scenario is depicted in Fig. 1, where the FD-AF relay network consists of BS, relay, and user; BS is equipped with a massive antenna and the relay and user have single antennas. It is assumed that the channel between BS and relay follows Rayleigh fading, and the channel between relay and user follows Nakagami-m fading. The simulation target is to select one antenna from the BS to complete transmission and verify the correctness of our theoretical analysis. As shown in Fig. 1, the parameter β increasing can bring about the improvement of the system performance, because the parameter β represents the strength of line-of-sight.

Figure 2 plots the PDF of end-to-end SNR results of the FD-AF relay network, where \( \beta =\alpha /{\tilde{\Omega}}_1=1,2 \) represents the line of sight level of direct link, (σ(R))2 = (σ(1))2 = (σ(2))2 = 0.1. As shown in Fig. 2, the proposed approximation method is effective when BS antenna number is high. Moreover, β has a certain influence on the approximation result: The gap between analytical results and simulation reduces gradually as BS antenna number and β increase. The parameter β represents the line of sight between transmitter and receiver; therefore, the parameter β is bigger and the system performance is better.

Fig. 2
figure 2

The PDF curve of end-to-end SNR

Figure 3 compares the system outage probability tendency with different parameters, where outage threshold R0 = 1bit/s/Hz. As shown in Fig. 3, the asymptotic curves drawn via Eq. (25) approach the simulation result as BS antenna number grows. Moreover, the system outage probability decreases as SNR increases, as well as β, the reason being that β represents the line of sight level with fixed \( {\tilde{\Omega}}_i\left(i=1,2\right) \), so SNR or β augment may result in an equivalent increase in SNR at the receiver.

Fig. 3
figure 3

The tendency of system outage probability with NB

Figure 4 shows the analytical results alongside the Monte Carlo simulations, which demonstrate the correctness of our outage probability analysis. We also found that insofar as the system outage probability is almost equivalent to the worst link outage probability, it is similar to the “bucket effect” in practice.

Fig. 4
figure 4

The system and link outage probability relationship

Figure 5 displays the tendency of asymptotic, analytical, and simulation system outage probability results as SNR increases. In particular, Fig. 5 reveals that the system outage probability curves coincide with increase in BS antennas; in effect, high SNR or transmit power has significant impact on the outage probability and other system performance factors. It is worth noting that system outage probability degrades considerably as the BS antenna or β decrease.

Fig. 5
figure 5

The tendency of outage probability with SNR

Figure 6 describes the correctness of the system average BER analytical results with Monte Carlo simulations. Several situations are compared in Fig. 6 to show that BS antenna number and channel parameter β have a consistent influence on the system’s average BER. Similarly, system average BER decreases dramatically as NBor β increase.

Fig. 6
figure 6

The tendency of average BER with SNR

6 Conclusion

The full-duplex system can achieve two times the spectrum gains compared to the half-duplex. In this paper, we proposed an antenna selection scheme for FD-AF relay networks based on max-min criterion. The closed-form outage probability and BER of FD-AF relay systems in flat Gamma fading channels also were derived completely. The performance analysis results revealed that the proposed scheme is superior to the conventional scheme. We also verified our results via Monte Carlo simulations.



Third Generation Partnership Project


Amplify and forward


Antenna selection


Bit error ratio


Base station


Cumulative distribution function


Channel state information

FD relay:

Full-duplex relay


Half duplex


Long-term evolution


Multiple-input multiple-output


Probability density function


Time division duplex


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This work was supported by the National Thirteen Five National Defense Fund under Grant 6140311030207; National Natural Science Foundation of China under Grants 61801170, 61501405, U1504619, U1404615, 61271018, 61801435, and 61671144; China Postdoctoral Science Foundation under Grant 2018M633351; Natural Science Foundation of Henan under Grant 162300410096 and 182102110401; Program for Science & Technology Innovation Talents in the University of Henan Province (17HASTIT025); Tibet National University Tibet cultural heritage development Collaborative Innovation Center 2018 tender twenty-first topic and Tibet Natural Science Foundation.

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BF J, YW, and HM are commonly derived the outage probability and BER expressions, especially the series and integrals. YQ L and GZ simulate the system performance and verify the correctness of the derivations. HW and LS modify the paper writing and smooth the words in the paper. All authors read and approved the final manuscript.

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Correspondence to Baofeng Ji.

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This appendix mainly derives the average BER expression in the paper as follows:

Based on the approximation expression, the equivalent channel CDF after BS antenna selection can be expressed as (28):

$$ {P}_{\gamma_e}(z)=1-\left[1-{\left(1-{e}^{-\frac{z}{{\tilde{\Omega}}_1}}\right)}^{N_B}\right]\cdot \frac{\Gamma \left(\alpha, \beta z\right)}{\Gamma \left(\alpha \right)} $$

Then, the \( {P}_{\gamma_e} \)can be calculated through binomial theorem as:

$$ {P}_{\gamma_e}(z)=1-\frac{\Gamma \left(\alpha, \beta z\right)}{\Gamma \left(\alpha \right)}+\sum \limits_{i=0}^{N_B}{C}_{N_B}^i\cdot {\left(-1\right)}^i\cdot {e}^{-\frac{iz}{{\tilde{\Omega}}_1}}\cdot \frac{\Gamma \left(\alpha, \beta z\right)}{\Gamma \left(\alpha \right)} $$

The general average BER mathematical expression can be defined as [30]:

$$ {P}_e\approx \frac{1}{2}{\int}_0^{\infty }{t}_1Q\left(\sqrt{2{t}_2z}\right){p}_{\gamma_e}(z) dz $$

where t1 and t2 are modulation coefficient and Q() is tail function of normal distribution and satisfies:

$$ Q(x)=\frac{1}{2\pi }{\int}_x^{\infty}\exp \left(-\frac{u^2}{2}\right) du $$

Therefore, Pe can be rewritten as:

$$ {\displaystyle \begin{array}{l}{P}_e\approx \frac{1}{2}{\int}_0^{\infty }{t}_1Q\left(\sqrt{2{t}_2z}\right){p}_{\gamma_e}(z) dz\\ {}=\frac{t_1\sqrt{t_2}}{4\sqrt{\pi }}{\int}_0^{\infty}\frac{e^{-{t}_2z}}{\sqrt{z}}{P}_{\gamma_e}(z) dz\\ {}=\frac{t_1\sqrt{t_2}}{4\sqrt{\pi }}{\int}_0^{\infty}\frac{e^{-{t}_2z}}{\sqrt{z}}\left[1-\frac{\Gamma \left(\alpha, \beta z\right)}{\Gamma \left(\alpha \right)}+\sum \limits_{i=0}^{N_B}{C}_{N_B}^i\cdot {\left(-1\right)}^i\cdot {e}^{-\frac{iz}{{\tilde{\Omega}}_1}}\cdot \frac{\Gamma \left(\alpha, \beta z\right)}{\Gamma \left(\alpha \right)}\right] dz\\ {}=\frac{t_1\sqrt{t_2}}{4\sqrt{\pi }}\frac{\beta^{\alpha}\Gamma \left(\alpha +\frac{1}{2}\right)}{{\left(\beta +{t}_2\right)}^{\alpha +\frac{1}{2}}\Gamma \left(\alpha +1\right)}{{}_2F}_1\left(1,\alpha +\frac{1}{2};\alpha +1;\frac{\beta }{\beta +{t}_2}\right)+\\ {}\frac{t_1\sqrt{t_2}}{4\sqrt{\pi }}\sum \limits_{i=0}^{N_B}{C}_{N_B}^i{\left(-1\right)}^i\frac{2{\beta}^{\alpha}\Gamma \left(\alpha +\frac{1}{2}\right)}{{\left(\beta +{t}_2+\frac{i}{{\tilde{\Omega}}_1}\right)}^{\alpha +\frac{1}{2}}\Gamma \left(\alpha \right)}{{}_2F}_1\left(1,\alpha +\frac{1}{2};\frac{3}{2};\frac{t_2+\frac{i}{{\tilde{\Omega}}_1}}{\beta +{t}_2+\frac{i}{{\tilde{\Omega}}_1}}\right)\end{array}} $$

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Ji, B., Li, Y., Meng, Y. et al. Performance analysis of two-way full-duplex relay with antenna selection under Nakagami channels. J Wireless Com Network 2018, 265 (2018).

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  • Full-duplex relay
  • Amplify-and-forward relay
  • Antenna selection
  • Outage probability
  • Bit error rate