Outage analysis for joint antenna and path selection decode-and-forward relay networks

Abstract

In this paper, we consider a multiple-input multiple-output (MIMO) cooperative communication system with single decode and forward (DF) relay. The system performance is analyzed by using joint antenna and path selection (JAPS) scheme over Nakagami-m fading channel. Then, we propose to use both JAPS and Sub-optimal transmit antenna selection strategy denoted by JAPS/maximum-ratio combining (JAPS/MRC) to improve the coding gain performance. For JAPS, a closed-form expression for the outage probability (OP) is derived. The upper bound of OP in high signal-to-noise ratio (SNR) regime is obtained. For JAPS/MRC a closed-form expression for the OP is also derived. The derived expressions are applicable with any number of antennas, different $m_i$ and distinct average SNRs between the paths. Simulation results are presented and show the validity of our theoretical analysis.

Introduction

Cooperative communication (CC) has obtained comprehensive attention due to its ability to extend the coverage area, enhance the system capacity and combat the performance degrading effects of the wireless fading channels [1]. Furthermore, the performance of CC can be improved when incorporated with multiple-input multiple-output (MIMO) technology [2], [3], [4], [5], but at the expense of cost and hardware complexity. One can use antenna selection (AS) to maintain many advantages of MIMO systems with lower hardware complexity [6], [7], [8]. In CC and at the relay, the decode-and-forward (DF) is a relaying protocol, in which, the relay first decodes the receiver's signal, re-encodes it, then forwards it to the destination [3], [8]. However, using DF protocol, erroneous relaying detection at the relay leads to diversity loss and performance degradation. In this paper, we focus on single relay employing DF where all nodes are equipped with multiple antennas.

Two sub-optimal antenna selection strategies exist in literature, either maximizing the signal-to-noise ratio (SNR) of the direct path (Sub₁) or the cooperative path (Sub₂). For a single relay, in [2], the asymptotic outage probability (OP) was derived for the two sub-optimal strategies with receive-MRC (R-MRC). Both DF and selective-DF (S-DF) relaying protocols are considered. In [9], closed-form OP expressions are derived for Sub₁ strategy for the two cases, i) transmit-receive antenna selection (TRAS), ii) transmit antenna selection (TAS)/R-MRC. For multiple relay network, joint antenna-relay-selection (JARS) with R-MRC, in [8] and [10], an exact closed-form OP without considering the direct link is derived. In [6], the authors considered a system with direct link and K MIMO relays using S-DF protocol. Closed-form OP expressions for relay selection are derived using Sub₁. Furthermore, the performance of CC with relay selection can be improved when we introduce buffers at the relay nodes [11], [12], [13], however this performance improvement increases packets delay.

A uniform antenna selection scheme for two-way relay channel (TWRC) network is proposed in [14]. The proposed scheme is based on maximizing the worse received SNR of the end users and is denoted by max-min (MM) criterion. In [15] the MM criterion is used for TWRC network for multiple MIMO relays. A phase-rotation MM relay selection scheme is proposed to improve the end-to-end symbol error rate (SER) performance. In [16], MM criterion and two other relay-selection strategies are proposed (max-sum (MS) and max-product (MP)) for non-coherent modulation to improve the bit error rate (BER) and the throughput while keeping hardware implementation easier. In addition, the MM criterion has been also used for downlink multi-users non orthogonal multiple access (NOMA) [17] and downlink cooperative multiple users NOMA [18], [19].

In CC system with a single relay, two paths are available the direct path and the cooperative path which can be seen as one-way relay channel. For path selection, the MM criterion can be used for path selection.¹ In addition, to overcome the MIMO complexity and DF diversity loss, joint AS and path selection can be used and denoted by joint antenna and path selection (JAPS) strategy [7], [20].

For JAPS strategy with a single relay, in [3], the analysis was from ergodic capacity perspective using orthogonal space-time block coding (OSTBC). In [20] using TRAS, both ergodic capacity and SER were analyzed. In [5], an asymptotic bound for SER is derived for any arbitrarily transmission scheme and fading channel. In [4], a scenario closer to practical using path selection and MIMO beamforming system is considered. In [7], the authors studied the SER for JARS strategies where the relays were equipped with a single antenna.

In [21] the authors investigated the performance of JAPS (similar to [20]) scheme as well as sub-optimal strategies schemes for an energy harvesting single antenna DF-relay ($N_r=1$). The exact analytical OP expressions are derived for the three schemes over Rayleigh fading channel.

Different from all the above literature discussed works, no works prior to this one have analyzed the OP of JAPS over Nakagami-m fading channel. Also, we have considered two schemes: the existing JAPS scheme as well as our proposed JAPS/MRC scheme both over the Nakagami-m fading channel. All source, relay and destination are equipped with multiple antennas. We take into account the direct link between the source and the destination. In this context, and view the importance of OP, we derive closed form expressions for the OP. Furthermore, we derive an upper bound at high SNR for OP where we extract the diversity order (DO). Then, we propose JAPS/MRC which uses JAPS and sub-optimal antenna selection strategy to improve the coding gain performance. A closed form expression for the OP is derived. Note that this strategy does not require two radio frequencies (RF) at the destination, since we combine two packets received in two time slots using the same RF chain. In addition between two nodes, we use both TRAS. Therefore, this work can be easily extended to transmit receive-MRC since the sum of gamma random variables is also gamma random variable [22].

The remainder of the paper is organized as follows. In Section 2, we present the system model. In Section 3, we derive the OP. Simulations and results are given and discussed in Section 4. We conclude this work in Section 5.

Notations: Let $N_s,N_r$ and $N_d$ be the number of antennas at the source, the relay and the destination, respectively. For simplicity, let $N_1=N_s{N}_d,N_2=N_s{N}_r$ and $N_3=N_r{N}_d$. We use s, r and d to denote the source, the relay and the destination, respectively. Similarly, we have $(uv)\in \{(sd),(sr),(rd)\}$. The total transmit power is $P_t$. Let $P_s=\mu P_t$ and $P_r=\nu P_t$ be the transmitted power at source and relay, respectively with $\mu +\nu =1$. Let $h_{u_i{v}_j}$ and $n_{u_i{v}_j}$ be the channel coefficient and the noise respectively, between the ith transmit antenna at u and the jth receive antenna at v. All $h_{u_i{v}_j}$ between u and v are identical independent and modeled as Nakagami-m (integer m) random variable. All $n_{u_i{v}_j}$ are complex additive white Gaussian noise (AWGN) with zero mean and variance $N_0$. The coefficient ${\varphi }_{x,y,z}$ function is given by [22, Eq. (8)]. Let $X\sim \mathcal{G}(m_i,{\beta }_i)$ be a random variable which follows gamma distribution with parameters ${\beta }_i$ and $m_i$. The probability density function (PDF) and cumulative distribution function (CDF) of X are

$$f_X(x)=\frac{m_i^{m_i}{\beta }_i^{m_i}}{\mathrm{\Gamma }(m_i)}{x}^{m_i-1}{e}^{-m_i{\beta }_{i}x}$$

$$F_X(x)=\frac{\gamma (m_i,m_i{\beta }_{i}x)}{\mathrm{\Gamma }(m_i)}.$$

Let $(m_1,{\beta }_1),(m_2,{\beta }_2)$ and $(m_3,{\beta }_3)$ be the channel parameters of source-destination, source-relay and relay-destination respectively. Also, we put $m_4=m_1$. For channel i where $i\in \{1,2,3\}$ (see Fig. 1), ${\zeta }_i$ and ${\sigma }_i$ are given in (3) (or see Appendix A). In addition $\widehat{{\sum }_{n_i,k_i}^{N_i}}$ is given in (3).

$$\widehat{\sum _{n_i,k_i}^{N_i}}=\sum _{n_i=0}^{N_i}\sum _{k_i=0}^{n_i(m_i-1)}\left(\begin{array}{c}{N}_i\\ n_i\end{array}\right){(-1)}^{n_i},{\zeta }_i={\acute{\beta }}_i+\sum _{\begin{array}{c}\hfill j=1\hfill \end{array}}^3{\acute{\beta }}_j{n}_j,{\sigma }_i=m_i+\sum _{\begin{array}{c}\hfill j=1\hfill \end{array}}^3{k}_j.$$