Performance Analyses of TAS/Alamouti-MRC NOMA in Dual-Hop Full-Duplex AF Relaying Networks

In [9], the authors have focused on design of precoding/detection structure for MIMO-NOMA system and investigated the impact of user pairing scheme to enlarge the performance gap as compared to the counterpart OMA. However, exploiting precoding based techniques has caused ideal channel state information (CSI) burden in wireless communications systems. On the contrary, space-time block coding (STBC) [10] does not require CSI, so it is more robust to CEEs when compared to the other MIMO techniques. Therefore, in [11], a downlink NOMA system, in which two coordinated BSs with single antenna perform distributed Alamouti-STBC to increase the reliability of cell-edge user, has been proposed and investigated through transmission rates. In [12], conventional Alamouti-STBC scheme has been applied to downlink multi-user NOMA system, where users are equipped with single antenna, and exact OP expression is obtained for any user over Nakagami-$m$ fading channels. In [13], the same system investigated in [12] has been generalized to cover all STBC schemes [14] by taking into account practical impairments such as CEE, feedback delay (FBD) and imperfect SIC. However, using multiple antennas at the transmitter and receiver ends increases hardware complexity and power consumption due to multiple radio-frequency (RF) chains. In order to overcome these drawbacks, antenna selection (AS) approach has been considered since it reduces the number of used RF chains as well as retaining many advantages of spatial diversity. Also, only partial CSI is required within the scope of AS. Thereby, several optimal and sub-optimal AS techniques have been proposed and investigated for OMA systems so far. In addition, some sub-optimal AS techniques such as max-max-max, max-min-max and maximum-channel-gain-based one have been addressed for MIMO-NOMA systems consisting of one BS and two users in [15]. In [16], by considering the similar system in [15], the authors have proposed a joint AS scheme, which ensures minimizing the OP by selecting the best antenna at the BS and optimal antennas at the two users, and obtained closed-form OP for Nakagami-$m$ fading channels. However, previously proposed AS schemes consider selection processes based on individual users. Therefore, the authors of [17] have proposed a novel AS scheme, which selects the best antenna of BS according to the majority decision of users, for single hop multi-user NOMA and investigated the OP performance over Nakagami-$m$ channels in the presence of CEE and FBD effects. In [18], by using the AS technique proposed in [17], transmit AS (TAS)/Alamouti-STBC scheme is investigated for single-hop MIMO-NOMA with practical impairments.

Besides multi-antenna schemes, there are many studies considering relaying techniques in NOMA network with/without MIMO in the literature due to the demands on expanding the coverage area. Accordingly, in [19], a downlink NOMA network based on coordinated direct and relay transmission, where the BS directly communicates with one user while a dedicated relay assists the other user, has been investigated. In [20], performance of a downlink cooperative NOMA network in which the BS serves multiple users through a dedicated amplify-and-forward (AF) relay has been analyzed in terms of OP over Nakagami-$m$ fading channels by also considering CEE effects. In [21], Alamouti-STBC scheme has been investigated in a cooperative decode-and-forward (DF) relaying network with direct link, where the BS communicates with only one user by adopting NOMA principle. On the other hand, AS techniques have also been investigated in cooperative NOMA networks. In [22], OP performance of a downlink cooperative NOMA network with AF relay, where the BS and multiple users apply TAS and maximal-ratio combining (MRC) techniques, respectively, has been analyzed over Nakagami-$m$ fading channels under CEE effects. In [23], the same system in [22] has been addressed in case of not only CEE but also imperfect SIC effects when the relay operates in variable and fixed gain modes. In [24], outage performance of TAS/Alamouti scheme is studied in dual-hop NOMA network for HD AF relay and two users by only simulations over Rayleigh fading channels.

Aforementioned studies within the scope of cooperative NOMA networks are based on half-duplex (HD) relaying technique. Although HD relaying increases reliability of the system, it reduces the spectral efficiency due to its principle of allocation of two orthogonal channels. Fortunately, to overcome this problem, full-duplex (FD) relaying has been proposed and regarded as a promising technique due to its ability of improving the spectral efficiency [25, 26, 27]. Since reliability of the system becomes crucial when the direct link between the BS and users does not exist due to huge obstacles or heavy shadowing, some studies have focused on exploiting benefits of the dedicated FD relays within the scope of cooperative NOMA [28, 29, 30, 31, 32, 33]. In [28], the authors propose and investigate a cooperative two-user NOMA network with/without direct link, where one user close to the BS acts as a DF relay switching between FD and HD mode to assist the other user far to the BS. To measure the level of performance, OP, ergodic rate and energy efficiency have been addressed in the case of Rayleigh fading channels. In [29], a cooperative two-user NOMA network with DF FD relay has been proposed and OP performance together with ergodic sum capacity have been investigated over Rayleigh fading channels. Particularly, in the investigated system, the BS transmits information directly to the relay and near user while the relay forwards decoded information to far and near users at the same time. In [30], a cooperative three-user NOMA network, where the BS communicates with two far users through a DF FD relay, has been considered and analyzed in terms of OP and ergodic sum capacity. In [31], exactly the same system investigated in [29] has been considered and analyzed over independent and identically distributed (i.i.d.) Nakagami-$m$ fading channels in terms of OP together with the ergodic rate, however the residual self-interference (SI) at the FD relay is assumed as Gaussian distribution. In [32], the same system in [29] has been addressed in point of AS problem by assuming that the BS and relay are equipped with multiple antennas. For performance criterion, closed-form OP and ergodic sum rate expressions are obtained. In addition, relay selection techniques have been investigated in two-user cooperative NOMA with multiple AF FD relays over Rayleigh fading channels in [33]. However, in [33], exact closed-form OP expressions could not be provided, and the analyses have been supported through lower bound and asymptotic approximations.

The studies on FD-NOMA network mentioned above in the literature review have demonstrated that the spectral efficiency of a cooperative NOMA system can be increased via FD relaying and also FD-NOMA outperforms the HD-NOMA counterpart under some certain conditions. In addition, using multiple antennas in NOMA has been shown to be quite efficient in improving the system performance. Moreover, given the significant advantages of TAS and Alamouti-STBC multiple antenna techniques mentioned in the literature review such as a reduced CSI burden together with benefits of spatial diversity and reduced RF complexity and power consumption, we believe that it is reasonably exciting to exploit the combination of them jointly. With the motivation of promising results gained from MIMO-NOMA and cooperative NOMA networks examined above, in this paper, our main focus has become investigating conventional TAS/Alamouti-STBC scheme in dual-hop AF FD multi-user power-domain NOMA network over i.i.d. Nakagami-$m$ fading channels in the presence of CEE and FBD effects. Particularly, in the considered system, the best two antennas at the BS are selected according to conventional TAS technique to apply Alamouti-STBC scheme while receivers of mobile users are assumed to combine received information by using MRC technique. The major contributions of this paper are highlighted as follows:

• •

  Unlike most of the existing studies carried out within cooperative FD-NOMA, we consider that the FD relay operates in AF mode and also residual SI link between antennas of the relay is exposed to fading effects. In addition, this paper is the first attempt that analyzes TAS/Alamouti scheme in cooperative NOMA literature and even assumes all channels undergo Nakagami-$m$ fading. It is worth noting that benefits of both the reduced RF chains and STBC have been brought to cooperative NOMA systems with the combination of TAS and Alamouti schemes.

• •

  In order to demonstrate the level of system performance, exact OP expression for any user is derived in single-fold integral through moment generating function (MGF) approach and simple lower bounds together with asymptotic expressions are also obtained to provide more meaningful insights into the OP. Thus, with the asymptotic analyses, the impacts of CEE, FBD and residual SI on the performance in asymptotic regime are provided more clearly.

• •

  Furthermore, test-bed implementation of the investigated system, which is the most sophisticated NOMA implementation in the literature, is conducted through USRP software defined radios (SDRs) to demonstrate its feasibility in a practical manner.

• •

  Numerical results of the investigated TAS/Alamouti-STBC FD-NOMA system are verified by Monte Carlo simulations and SDR-based real-time tests and compared to both HD-NOMA and FD-OMA counterparts. Results demonstrate that the quality of SI cancellation in the FD relay is mostly effective in OP performances of the users with lower power levels whose performances are also highly effected by the number of antennas at the BS.

Bold lowercase letter and $\|\cdot\|_{F}$ denote vectors and the Frobenius norm of a vector, respectively. $\mathbb{CN}(0,\sigma^{2})$ indicates the complex Gaussian distribution with zero mean and variance of $\sigma^{2}$. While $E\left[\cdot\right]$ represents the expectation operator, $Pr(\cdot)$ denotes the probability of any event.

The reminder of the paper is given as follows. The investigated system and channel statistics are introduced in Section II. The exact OP together with lower bound and asymptotic analyses are performed in Section III. Test-bed implementation of the investigated system is presented in Section IV. Finally, Sections V and VI provide numerical results and conclusions, respectively.

In the network, channel coefficient vectors related to $S-R$, $R-U_{l}$ and $R-R$ links are denoted by $\mathbf{h}_{SR}=\left\{h_{SR}^{i}\right\}_{n_{B}\times 1}$, $\mathbf{h}_{RU_{l}}=\left\{h_{RU_{l}}^{j}\right\}_{1\times n_{R}}$ and $\mathbf{h}_{RR}=\left\{h_{RR}^{t}\right\}_{2\times 1}$ ($1\leq i\leq n_{B}$ and $1\leq j\leq n_{R}$) whose elements are modeled as i.i.d. Nakagami-$m$ distribution with powers $\Omega_{SR}=E\left[|h^{i}_{SR}|^{2}\right]=d_{SR}^{-\eta}$, $\Omega_{RU_{l}}=E\left[|h^{j}_{RU_{l}}|^{2}\right]=d_{RU_{l}}^{-\eta}$ and $\Omega_{RR}=E\left[|h^{t}_{RR}|^{2}\right]=\alpha P_{R}^{\mu-1}$, respectively. Here, $d_{SR}$, $d_{RU_{l}}$ and $\eta$ represent normalized distances of $S-R$ and $R-U_{l}$ links and path loss exponent, respectively. Also, we assume that the relay suffers from residual SI effect resulting from active and/or passive SI cancellation techniques which are modeled as in [25] and [26]. Therefore, constants $\alpha$ and $\mu$ denote the quality of SI cancellation process ($0\leq\mu\leq 1$, $\alpha>0$).

In the training period before the actual transmission, the relay is assumed to estimate channel gains of both $S-R$ and $R-U_{l}$ links for the determination of indices belonging to be selected antennas of the BS and power coefficients to be allocated to users, respectively. In this context, we consider that CEE and FBD exist in both hops since they are inevitable in case of practical implementations. Note that there is no CEE and/or FBD effects on SI link since channel estimation process is not carried out and only SI cancellation techniques are used. In the first hop, since two antennas of the BS are selected according to optimum TAS technique at the relay, squares of the estimated channel gains are sorted in descending order as $|\hat{h}^{1}_{SR}|^{2}>|\hat{h}^{2}_{SR}|^{2}>\cdots>|\hat{h}^{n_{B}}_{SR}|^{2}$. Then, two antennas provide the highest channel gains are selected to maintain the actual transmission. $S-R$ channel coefficient vector resulting from antenna selection process is denoted by $\hat{\tilde{\mathbf{h}}}_{SR}=\begin{bmatrix}\hat{h}^{1}_{SR}\\ \hat{h}^{2}_{SR}\end{bmatrix}$. By taking into account the CEE and FBD effects, the channel coefficient vector corresponding to the first hop is re-expressed as $\mathbf{\tilde{h}}_{SR}=\rho_{SR}\hat{\tilde{\textbf{h}}}^{\tau}_{SR}+\boldsymbol{\varepsilon}_{fbd,SR}+\boldsymbol{\varepsilon}_{est,SR}$, where $\hat{\mathbf{h}}$ and $\hat{\mathbf{h}}^{\tau}$ denote the erroneously estimated channel vector and its delayed version. $\boldsymbol{\varepsilon}_{est,SR}\sim\mathbb{CN}(0,\sigma_{est,SR}^{2})$ and $\boldsymbol{\varepsilon}_{fbd,SR}\sim\mathbb{CN}(0,\sigma_{fbd,SR}^{2})$ denote CEE and FBD vectors whose entries’ variances are found by $\sigma_{est,SR}^{2}=\Omega_{SR}-\hat{\Omega}_{SR}$ and $\sigma_{fbd,SR}^{2}=\left(1-\rho_{SR}^{2}\right)\hat{\Omega}_{SR}$, respectively. $\rho_{X}=J_{0}\left(2\pi f_{D}\tau\right)\left(0<\rho<1\right)$ is the correlation coefficient corresponding to link $X$ where $J_{0}(\cdot)$, $f_{D}$ and $\tau$ represent zero-order Bessel function of the first kind, maximum Doppler frequency and time delay, respectively [34, 35]. Then, CEE and FBD vectors can be represented by a new random variable as $\boldsymbol{\varepsilon}_{SR}=\boldsymbol{\varepsilon}_{fbd,SR}+\boldsymbol{\varepsilon}_{est,SR}$ with a variance of $\sigma_{SR}^{2}=\sigma_{fbd,SR}^{2}+\sigma_{est,SR}^{2}$ [36, 37]. On the other hand, in the second hop, the channel coefficient vector of $R-U_{l}$ link is expressed as $\mathbf{h}_{RU_{l}}=\rho_{RU_{l}}\hat{\mathbf{h}}^{\tau}_{RU_{l}}+\boldsymbol{\varepsilon}_{fbd,l}+\boldsymbol{\varepsilon}_{est,l}$. Similar to the first hop, $\boldsymbol{\varepsilon}_{est,l}\sim\mathbb{CN}(0,\sigma_{est,l}^{2})$ and $\boldsymbol{\varepsilon}_{fbd,l}\sim\mathbb{CN}(0,\sigma_{fbd,l}^{2})$ denote CEE and FBD vectors with variances of $\sigma_{est,l}^{2}=\Omega_{RU_{l}}-\hat{\Omega}_{RU_{l}}$ and $\sigma_{fbd,l}^{2}=\left(1-\rho_{RU_{l}}^{2}\right)\hat{\Omega}_{RU_{l}}$. Afterwards, CEE and FBD vectors can be represented by $\boldsymbol{\varepsilon}_{RU_{l}}=\boldsymbol{\varepsilon}_{fbd,l}+\boldsymbol{\varepsilon}_{est,l}$ with a variance of $\sigma_{RU_{l}}^{2}=\sigma_{fbd,l}^{2}+\sigma_{est,l}^{2}$.

In the first hop, the signal received by the relay in the $n$th time interval is expressed as

where $\mathbf{w}_{R}[n]=\left\{w_{R}^{t}\right\}_{2\times 1}$ denotes noise vector at the relay whose entries are distributed as $\mathbb{CN}(0,\sigma_{R}^{2})$. While $\mathbf{s}_{R}[n]=\mathcal{G}\mathbf{y}_{R}[n-D]=\left\{\mathcal{G}y_{R}^{t}\right\}_{2\times 1}$ represents the signal vector transmitted from $R$ to $U_{l}$, $D$ and $\mathcal{G}$ denote processing delay and amplifying factor of the FD relay. Also, $(\bullet)$ is the component-wise product of two vectors with the same dimension. Then, the received signal vector $\mathbf{y}_{SRU_{l}}\left[n\right]=\left\{y_{SRU_{l}}^{t,j}\right\}_{2\times n_{R}}$ by the $l$th user can be written as

where $\mathbf{w}_{l}[n]=\left\{w_{l}^{t,j}\right\}_{2\times n_{R}}$ denotes noise vector at the $l$th user whose entries are distributed as $\mathbb{CN}(0,\sigma_{l}^{2})$. Without loss of generality, $\sigma_{R}^{2}=\sigma_{l}^{2}=\sigma^{2}$ is assumed to be mathematically tractable. By substituting $\mathbf{s}_{R}[n]$ into (2), the received signal vector at the $l$th user can be rewritten as

In (3), the first line comprises both the desired signal related to $U_{l}$ and interference signals coming from other users. Then, received signals by mobile users are combined according to MRC technique. With the help of (1), amplifying factor of the FD relay can be determined as

where $P_{S}$ is the transmit power of the BS. However, for mathematical simplicity, $P_{S}=P_{R}=P$ is assumed in the continuation of the analysis.

Without loss of generality, we assume that power allocation coefficients ($a_{l}$) of mobile users are determined according to statistics of $R-U_{l}$ links since direct link between the BS and mobile users can not be established due to the poor channel conditions. Therefore, in order for the BS to determine the power allocation coefficients of users, the relay estimates the effective channel gains of $R-U_{l}$ link and sort as $\|\hat{\mathbf{h}}_{RU_{1}}\|_{F}^{2}\leq\|\hat{\mathbf{h}}_{RU_{2}}\|_{F}^{2}$ $\cdots\leq\|\hat{\mathbf{h}}_{RU_{L}}\|_{F}^{2}$. Then, the relay transmits the ordering to both BS and users through feedback channel. The BS determines power levels as $a_{1}>a_{2}>\cdots>a_{L}$ (subjected to $\sum_{l=1}^{L}a_{l}=1$) with the help of the ordering. By using the decoding method of STBC as in [38] and considering SIC technique is perfectly carried out at mobile users, instantaneous SINR that $U_{l}$ can decode the signal of $U_{k}$ ($k\leq l$) is expressed as

where $\bar{\gamma}=P/\sigma^{2}$ is the average SNR, and $A\stackrel{{\scriptstyle\triangle}}{{=}}\|\hat{\tilde{\mathbf{h}}}^{\tau}_{SR}\|_{F}^{2}$, $B\stackrel{{\scriptstyle\triangle}}{{=}}\|\hat{\mathbf{h}}^{\tau}_{RU_{l}}\|_{F}^{2}$ and $C\stackrel{{\scriptstyle\triangle}}{{=}}|h_{RR}|^{2}$ definitions are used for simplicity. In addition, constant variables represented by notation $\vartheta$ are given below:

By assuming $\left\{\gamma_{SRU_{k\rightarrow l}}>\gamma_{th,k}\right\}$ as the event that $U_{l}$ can detect the message corresponding to itself or $U_{k}$ ($1\leq k\leq l$), where $\gamma_{th,k}$ represents the targeted threshold SINR of $U_{k}$, the OP at $U_{l}$ is formulated as

Then, with the help of (5), the event $\left\{\gamma_{SRU_{k\rightarrow l}}>\gamma_{th,k}\right\}$ can be expressed as

which is obtained under the condition of $a_{k}-\gamma_{th,k}\sum_{t=k+1}^{L}a_{t}>0$ and the notation $\delta_{k}\stackrel{{\scriptstyle\triangle}}{{=}}\frac{\gamma_{th,k}}{\bar{\gamma}(a_{k}-\gamma_{th,k}\sum_{t=k+1}^{L}a_{t})}$ is used for mathematical tractability. If (8) is substituted into (7), the OP expression corresponding to $U_{l}$ can be written as

which is obtained by defining $\delta_{l}^{{\dagger}}=\underset{1\leq k\leq l}{\max}\left\{\delta_{k}\right\}$. By solving the probability problem given by (9), OP of $U_{l}$ can be mathematically expressed as

where $a=\frac{2(\vartheta_{2}\bar{\gamma}y+\vartheta_{3}\bar{\gamma}z+\vartheta_{4}\bar{\gamma}^{2}yz+\vartheta_{5})\delta_{l}^{{\dagger}}}{\bar{\gamma}(y-2\vartheta_{1}\delta_{l}^{{\dagger}})}$. In (10), $f_{X}(\cdot)$ and $f_{X}^{(l)}(\cdot)$ are used to represent the probability density functions (PDFs) of a random variable $X$ and its $l$th order statistic, respectively. The following theorem provides the exact OP related to the $l$th user.

Theorem 1: By solving (10), the exact OP for the $l$th user can be derived as

where

In (11), $K_{v}(\cdot)$ denotes the $v$th-order modified Bessel function of the second kind [39, eq.(8.407.1)] and $\pi_{l}(z)=z+\frac{2\vartheta_{1}\vartheta_{2}\bar{\gamma}\delta_{l}^{{\dagger}}+\vartheta_{5}}{\vartheta_{3}\bar{\gamma}+2\vartheta_{1}\vartheta_{4}\bar{\gamma}^{2}\delta_{l}^{{\dagger}}}$.

Proof: Please see Appendix A.

Unfortunately, there is no closed form solution of the integral $\Phi_{l}(z)$ in (11). However, an approximated form of (11) can be derived by upper-bounding the SINR given in (5). By applying some mathematical manipulations, (9) can be approximated as

where a new random variable $W\stackrel{{\scriptstyle\triangle}}{{=}}\bar{\gamma}A/\left(\bar{\gamma}C+\vartheta_{4}^{\prime}\right)$ is defined for mathematical tractability. Also, $\vartheta_{1}^{\prime}=\dfrac{\bar{\gamma}}{2}\dfrac{\sigma_{RU_{l}}^{2}}{\rho_{RU_{l}}^{2}}+\dfrac{1}{\rho_{RU_{l}}^{2}}$, $\vartheta_{2}^{\prime}=\dfrac{1}{\rho_{SR}^{2}}$, $\vartheta_{3}^{\prime}=\dfrac{\bar{\gamma}\sigma_{RU_{l}}^{2}+1}{\rho_{SR}^{2}\rho_{RU_{l}}^{2}}$ and $\vartheta_{4}^{\prime}=\dfrac{\bar{\gamma}^{2}}{2}\sigma_{SR}^{2}+1$. Then, the left hand side of the argument of $Pr(\cdot)$ can be upper-bounded by using the inequality $xy/(x+y)\leq\min(x,y)$. The right hand side of this property also provides the lower-bound for the OP as

where $F_{X}(\cdot)$ and $\overline{F}_{X}(\cdot)$ represent the cumulative distribution function (CDF) of a random variable $X$ and its complementary, respectively. Also, $F_{X}^{(l)}(\cdot)$ denotes the $l$th order statistic of $X$. Finally, if the complementaries of CDF expressions provided in the following theorem are substituted into (13), lower-bound of the OP for $l$th user can be obtained in closed-form.

Theorem 2: The CDF expressions of random variables predefined as $W$ and $B$ which also denote SINR and SNR of the first and second hops can be derived respectively as

where $\widetilde{\widetilde{\sum}}\equiv\sum_{r=0}^{n_{B}-2}\sum_{n=0}^{r(m_{SR}-1)}\sum_{t=0}^{m_{SR}-1}\sum_{t_{1}=1}^{T}\sum_{t_{2}=1}^{\alpha_{t_{1}}}\sum_{t_{4}=0}^{t_{2}-1}\sum_{t_{5}=0}^{t_{4}}$ and $\widetilde{\widetilde{\widetilde{\sum}}}\equiv\sum_{k=0}^{L-l}\sum_{p=1}^{l+k}\sum_{k_{1}=0}^{p(m_{RU_{l}}n_{R}-1)}$.

On the other hand, in case of ideal conditions, $\vartheta_{1}^{\prime}=\vartheta_{2}^{\prime}=\vartheta_{3}^{\prime}=\vartheta_{4}^{\prime}=1$ and the random variable $W$ can be approximated as $W\simeq A/C$. Therefore, $F_{W}(x)$ can be obtained for the ideal case as

where $\widetilde{\widetilde{\sum}}\equiv\sum\limits_{r=0}^{n_{B}-2}\sum\limits_{n=0}^{r(m_{SR}-1)}\sum\limits_{t=0}^{m_{SR}-1}\sum\limits_{t_{1}=1}^{T}\sum\limits_{t_{2}=1}^{\alpha_{t_{1}}}\sum\limits_{t_{4}=0}^{t_{2}-1}$. Also, the CDF of random variable $B$ remains the same for ideal case.

Proof: Please see Appendix B.

In this subsection, in order to provide more meaningful insights into the system performance, we investigate the OP in high SNR region by deriving simple theoretical expressions including diversity order and array gain metrics. The analyses are carried out in the following two subsections.