System Outage Probability of PS-SWIPT Enabled Two-Way AF Relaying with Hardware Impairments

As illustrated in Fig. 1, we consider a SWIPT based two-way AF relay network with TDBC, where the energy-constrained relay $R$ harvests energy from the RF signal via a PS protocol and adopts the “harvest-then-forward” scheme to assist the information interchange between the two terminals $S_{a}$ and $S_{b}$ [13, 14]. A direct link exists between the two terminals. All involved nodes are equipped with a single antenna and work in the half-duplex mode. Each transmission block $T$ is divided into three phases [24, 25]: two broadcast (BC) phases ($2T/3$) and one relay (RL) phase ($T/3$), as shown in Fig. 2. In the first BC phase, terminal $S_{a}$ broadcasts its own signal $x_{a}$ to relay $R$ and terminal $S_{b}$. In the second BC phase, terminal $S_{b}$ transmits its signal $x_{b}$ to relay $R$ and terminal $S_{a}$. After receiving the signal from $S_{i}$ ($i\!=\!a$ or $b$), relay $R$ splits the received power into two parts according to a PS ratio, i.e., a $\beta$ $\left(\beta\in(0,1)\right)$ portion of the received power for energy harvesting (EH) and the rest for information processing (IP). In the RL phase, relay $R$ amplifies the received signals and broadcasts them to the two terminals $S_{a}$ and $S_{b}$ using all the harvested energy.

All the channels are quasi-static, reciprocal and subject to independent but non-identically distributed Nakagami-$m$ fading. Specifically, the channel fading coefficient between $S_{i}$ and $S_{j}$ is denoted by $h_{ij}\sim{\rm{Naka}}(m_{d},\Omega_{d})$, with $i,j\in\left\{{a,b}\right\},\;i\neq j$. The channel fading coefficient of the $S_{i}$ to $R$ link is denoted by $h_{ir}\sim{\rm{Naka}}(m_{i},\Omega_{i})$. Since $h_{ij}$ and $h_{ir}$ follow the Nakagami distribution, the corresponding channel gains $|{h_{ij}}{|^{2}}$ and $|{h_{ir}}{|^{2}}$ follow the gamma distribution [26], and their PDF and CDF are expressed as

where $V\in\left\{{|{h_{ij}}{|^{2}},|{h_{ir}}{|^{2}}}\right\}$, $m$ and $\theta$ denote the shape parameter and the scale parameter of the variable $V$, respectively. Note that when $V{\rm{=}}|{h_{ij}}{|^{2}}$ $\left({|{h_{ir}}{|^{2}}}\right)$, $m={m_{d}}$ $\left({{m_{i}}}\right)$ and $\theta{\rm{=}}{\Omega_{d}}/{m_{d}}$ $\left({{\Omega_{i}}/{m_{i}}}\right)$ respectively.

In the first and second BC phases, $S_{a}$ and $S_{b}$ broadcast the information signal $x_{a}$ and $x_{b}$, respectively. For the direct link, the received signal at $S_{j}$ from $S_{i}$ ($i,j\in\left\{{a,b}\right\},i\neq j$), under the presence of HIs, is expressed as

where ${x_{i}}$ is the unit-power signal transmitted by $S_{i}$; $P_{i}$ denotes the transmit power of $S_{i}$; ${\tau_{ti}}\sim\mathcal{CN}\left({0,k_{1}^{2}}\right)$ denotes the hardware distortion noise caused by the transmitter of $S_{i}$ and ${\tau_{rj}}\sim\mathcal{CN}\left({0,k_{2}^{2}{P_{i}}|{h_{ij}}{|^{2}}}\right)$ denotes the hardware distortion noise caused by the receiver of $S_{j}$; ${{n_{ij}}}\sim\mathcal{CN}\left({0,\sigma_{ij}^{2}}\right)$ is the additive white Gaussian noise (AWGN) at $S_{j}$. Hereby, the parameters $k_{1}$ and $k_{2}$ characterize the HIs levels of the transmitter and receiver respectively [20]. Note that the product term in (3), ${\sqrt{{P_{i}}}{h_{ij}}{\tau_{ti}}}$, follows a Gaussian distribution¹¹1Under the assumption of quasi-static fading channels, the channel fading coefficient $h_{ij}$ is constant in each transmission block [20]. Meanwhile, the hardware distortion noise $\tau_{ti}$ caused by the transmitter of $S_{i}$ follows a Gaussian distribution with zero mean and variance $k^{2}_{1}$ [23, 21]. Thence, ${\sqrt{{P_{i}}}{h_{ij}}{\tau_{ti}}}\sim\mathcal{CN}\left({0,k_{1}^{2}{P_{i}}|{h_{ij}}{|^{2}}}\right)$. with zero mean and variance $k_{1}^{2}{P_{i}}|{h_{ij}}{|^{2}}$. For simplicity, we assume that $P_{a}=P_{b}=P_{o}$ and $\sigma_{ab}^{2}=\sigma_{ba}^{2}={\sigma^{2}}$ [13].

Based on (3), the SNDR of the direct link at $S_{j}$ is given as

where $\rho=\frac{{{P_{o}}}}{{{\sigma^{2}}}}$ denotes the input SNR.

As shown in Fig. 3, the received signal at $R$ from $S_{i}$ ($i\!=\!a$ or $b$), under the presence of HIs, can be written as

where ${n_{ir_{1}}}\sim\mathcal{CN}\left({0,\sigma_{ir_{1}}^{2}}\right)$ denotes the antenna noise at $R$, ${\tau_{ti}}\sim\mathcal{CN}\left({0,k_{1}^{2}}\right)$ denotes the hardware distortion noise caused by the transmitter of $S_{i}$.

Following the PS protocol, the received signal ${y_{ir}}$ is divided into two parts: $\sqrt{\beta}{y_{ir}}$ for EH and $\sqrt{1-\beta}{y_{ir}}$ for IP. Thus, the total harvested energy ²²2For analytical tractability, this work considers the linear EH model [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 21, 22, 23] instead of non-linear EH model [27, 28, 29]. Please note that some non-linear EH models can be decomposed into piecewise linear EH models, where each segment of the EH model is linear and the results obtained based on linear EH models can be applied to each segment [30, 31]. Combining the derived results in this work and the Law of Total Probability, the system outage probability under the non-linear EH model can be obtained by following [30, 31]. from both terminals can be expressed as [22]

where $\eta\in(0,1)$ denotes the energy conversion efficiency at relay $R$. Here we ignore the energy harvested from the hardware distortion noise or the AWGN as it is much smaller than that from the RF signals transmitted by the two terminals [23, 21].

Based on the “harvest-then-forward” scheme, the transmit power of $R$ is given as

Furthermore, the received signal for IP is written as

where ${\tau_{rr}}\sim\mathcal{CN}\left({0,k_{2}^{2}(1-\beta){P_{o}}\left({|{h_{ar}}{|^{2}}+|{h_{br}}{|^{2}}}\right)}\right)$ denotes the hardware distortion noise³³3In practice, the majority of the distortion noises at receiver is considered to be caused after the down-conversion process of the received antenna signal to baseband. Considering the complexity of theoretical analysis, the distortion noises cased by receiving process are neglected. caused by the receiver of $R$, ${n_{ir}}\sim\mathcal{CN}\left({0,\sigma_{ir}^{2}}\right)$ for $i\in\left\{{a,b}\right\}$ denotes the AWGN at $R$.

In the RL phase, the relay firstly amplifies ${y_{\rm{R}}^{\rm{IP}}}$ using an amplification gain $G$ and broadcasts the amplified signal to both terminals. The received signal at the node $S_{i}$ from the relay can be written as

where $G\approx\sqrt{\frac{{{P_{r}}}}{{\left({1-\beta}\right)\left({1+k_{1}^{2}+k_{2}^{2}}\right){P_{o}}\left({|{h_{ar}}{|^{2}}+|{h_{br}}{|^{2}}}\right)}}}$ [23], ${\tau_{tr}}\sim\mathcal{CN}\left({0,k_{1}^{2}}\right)$ denotes the distortion noise caused by the transmitter of $R$, ${\tau_{ri}}\sim\mathcal{CN}\left({0,k_{2}^{2}{P_{r}}\left({|{h_{ri}}{|^{2}}}\right)}\right)$ denotes the distortion noise caused by the receiver of $S_{i}$ and ${n_{ri}}\sim\mathcal{CN}\left({0,\sigma_{ri}^{2}}\right)$ denotes the AWGN at $S_{i}$. For simplicity, we assume that $\sigma_{ar}^{2}=\sigma_{br}^{2}=\sigma_{ra}^{2}=\sigma_{rb}^{2}={\sigma^{2}}$ [13].

After self-interference cancellation, the SNDR of the relaying link⁴⁴4Our derivations under the TDBC protocol can be applied to SWIPT based TWRNs with MABC, because under the MABC protocol, the SNDR of the relaying link at $S_{i}$ can be written as ${\gamma_{ri}}=\frac{{{I_{1}}{\rho}|{h_{ir}}{|^{2}}|{h_{jr}}{|^{2}}}}{{{I_{2}}{\rho}|{h_{ir}}{|^{2}}\left({|{h_{ar}}{|^{2}}+|{h_{br}}{|^{2}}}\right)+{I_{3}}|{h_{ir}}{|^{2}}/2+1}}$. at $S_{i}$ can be calculated as

with

Based on (4) and (10), the end-to-end SNDRs after selection combination at $S_{a}$ and $S_{b}$ can be written respectively as

Note that ${\gamma_{ab}=\gamma_{ba}}$ due to the reciprocity of the $S_{a}$ to $S_{b}$ channel.

The considered network will be in outage when either the transmission rate $\rm R_{a}$ from $S_{a}$ to $S_{b}$ or the transmission rate $\rm R_{b}$ from $S_{b}$ to $S_{a}$ is less than a given threshold $\rm R_{th}$. Thus, the system outage probability⁵⁵5From (6), we can see that relay $R$ may harvest insufficient energy $E_{b}$ if the two terminals transmit at a low power level. According to (7)-(10), this will lead to a low SNDR at both terminals and a high system outage probability., $\mathbb{P}_{\rm out}$, can be expressed as

where ${\gamma_{th}}={2^{{{3{{\rm{R}}_{{\rm{th}}}}}\mathord{\left/{\vphantom{{3{{\rm{R}}_{{\rm{th}}}}}T}}\right.\kern-1.2pt}T}}}-1$ denotes the SNDR threshold and ${\rm{R}}_{i}=\frac{T}{3}{\log_{2}}\left({1+{\gamma_{i}}}\right)$, $i\in\{a,b\}$.

Define $|{h_{ar}}{|^{2}}=X$, $|{h_{br}}{|^{2}}=Y$ and $|{h_{ab}}{|^{2}}=Z$. Then the system outage probability $\mathbb{P}_{\rm out}$ can be re-expressed as

Based on (1) and (2), we derive the four probabilities on the right-hand side of (16) to obtain the closed-form expression for the system outage probability $\mathbb{P}_{\rm out}$, respectively. The first term, $\mathbb{P}_{1}$, can be calculated as

where ${\mathbb{P}_{1}^{1}}$ is given by

The second term, $\mathbb{P}_{2}$ can be calculated as

where ${\mathbb{P}_{2}^{1}}$ is shown in (22) at the top of this page.

Similarly, the third term, $\mathbb{P}_{3}$, can be calculated as

where ${\mathbb{P}_{3}^{1}}$ is given in (24) at the top of this page.

The fourth term, $\mathbb{P}_{4}$, can be calculated as

In (24), $\mathbb{P}_{4}^{1}$ is shown in (26) at the top of the this page; $\mathbb{P}_{4}^{*}$ equals either $\mathbb{P}^{2}_{4}$ or $\mathbb{P}^{3}_{4}$, as shown respectively in (III-A) and (III-A) at the top of the next page. $\mathbb{P}^{2}_{4}$ corresponds to the case that the quartic function (C.3) in Appendix C have only one real root, and $\mathbb{P}^{3}_{4}$ corresponds to the case that the quartic function (C.3) have three different positive real roots.