Performance Analysis on Two-Way Relay System with Co-Channel Interference

Abstract

In this paper, we investigate the performance of two-way relay system with co-channel interference in a Rayleigh fading environment. Due to the mathematical intractability of original expression of signal-to-interference and noise ratio (SINR) induced by the bi-directional forwarding, a couple of effective bounds are derived for important performance metrics such as outage probability and average bit error rate of modulation, with application of diverse physical layer network coding (PNC) schemes of two time slot (2TS) and three time slot (3TS). Some asymptotic solutions are also proposed to intuitively exhibit the trends of performance in high SINR regime. We demonstrate that the performance bounds have practical meaning corresponding to different forwarding capability of the relay. Numeric simulations validate our analysis by showing that the theoretic bounds match well with simulation results. Additionally, the effect of distance between relay and sources is discussed in our interference scenario, as well as the impact of relay’s power allocation factor in 3TS PNC scheme.

Appendices

Appendix A: Proof of Lemma 1

The sum $||\mathbf Y ||_1=\sum _{i=1}^n Y_i$ follows a generalized chi-square distribution of which the PDF is (see [29])

$$\begin{aligned} f_{||\mathbf Y ||_1}(u)=\sum _{k=1}^{r(\bar{\mathbf{Y }})} \sum _{l=1}^{v_k(\bar{\mathbf{Y }})} \frac{\varphi _{k,l}(\bar{\mathbf{Y }})}{\Gamma (l)\bar{Y} _{[k]}^l}u^{l-1}\exp \left(-\frac{u}{\bar{Y}_{[k]}}\right) \end{aligned}$$

(45)

where the parameters are already introduced in the remarks of Lemma 1. Thus the final CDF in Lemma 1 is calculated as follows:

$$\begin{aligned} F_Z(z)&= \Pr (X<z(w+||\mathbf Y ||_1)) \nonumber \\&= \int \limits _0^{+\infty }F_X\left(z(w+u)\right)f_{||\mathbf Y ||_1}(u)\,\text{ d}u \nonumber \\&= \int \limits _0^{+\infty }\left[1-\exp \left\{ -\frac{z(w+u)}{\bar{X}}\right\} \right] \sum _{k=1}^{r(\bar{\mathbf{Y }})}\sum _{l=1}^{v_k(\bar{\mathbf{Y }})} \frac{\varphi _{k,l}(\bar{\mathbf{Y }})}{\Gamma (l)\bar{Y}_{[k]}^l}u^{l-1}\exp \left(-\frac{u}{\bar{Y}_{[k]}}\right)\,\text{ d}u \nonumber \\ \end{aligned}$$

(46)

$$\begin{aligned}&= 1-\sum _{k=1}^{r(\bar{\mathbf{Y }})}\sum _{l=1}^{v_k(\bar{\mathbf{Y }})}\varphi _{k,l}(\bar{\mathbf{Y }}) \left(1+\frac{\bar{Y}_{[k]}}{\bar{X}}z\right)^{-l} \exp \left(-\frac{w}{\bar{X}}z\right) \end{aligned}$$

(47)

Appendix B: Proof of Theorem 1

According to (18), we have

$$\begin{aligned} \mathcal P _\mathcal{O ,A}^{\tau \mathrm TS }(\gamma _\mathrm{th ,A})&> \Pr \{\gamma _{A,\mathrm ub }^{\tau \mathrm TS }<\gamma _\mathrm{th ,A}\} \\&= 1-\Pr \{\gamma _{A,\mathrm ub }^{\tau \mathrm TS }\ge \gamma _\mathrm{th ,A}\} \\&= 1-\Pr \left\{ \frac{c\rho _A}{\sigma _n^2+||\mathbf Q _A^{\tau \mathrm TS }||_1}\ge \gamma _\mathrm{th ,A}\right\} \cdot \Pr \left\{ \frac{P_B^{\tau \mathrm TS }}{N_{R,A}^{\tau \mathrm TS }+||\mathbf Q _R^{\tau \mathrm TS }||_1}\ge \gamma _\mathrm{th ,A}\right\} \end{aligned}$$

Utilizing Lemma 1, the lower bound of outage is immediately obtained as (23). On the other hand,

$$\begin{aligned}&\mathcal P _\mathcal{O ,A}^{\tau \mathrm TS }(\gamma _\mathrm{th ,A}) \\&\quad \approx 1-\Pr \{\gamma _{A,\mathrm lb }^{\tau \mathrm TS } \ge \gamma _\mathrm{th ,A}\} \\&\quad =1-\Pr \left\{ \frac{P_A^{\tau \mathrm TS }}{N_A^{\tau \mathrm TS } +||\mathbf Q _{R,A}^{\tau \mathrm TS }||_1}\ge \gamma _\mathrm{th ,A}, \frac{P_B^{\tau \mathrm TS }}{N_{R,A}^{\tau \mathrm TS } +||\mathbf Q _{R,A}^{\tau \mathrm TS }||_1} \ge \gamma _\mathrm{th ,A}\right\} \\&\quad =1-\int \limits _0^{+\infty }\Pr \{P_A^{\tau \mathrm TS }\ge \gamma _\mathrm{th ,A}(N_A^{\tau \mathrm TS }+y)\} \cdot \Pr \{P_B^{\tau \mathrm TS }\ge \gamma _\mathrm{th ,A} (N_{R,A}^{\tau \mathrm TS }+y)\} f_{||\mathbf Q _{R,A}^{\tau \mathrm TS }||_1}(y)\,\mathrm d y \\&\quad = 1-\int \limits _0^{+\infty }\exp \left(-\frac{\gamma _\mathrm{th ,A} (N_A^{\tau \mathrm TS }+y)}{\bar{P}_A^{\tau \mathrm TS }}\right) \cdot \exp \left(-\frac{\gamma _\mathrm{th ,A}(N_{R,A}^{\tau \mathrm TS }+y)}{\bar{P}_B^{\tau \mathrm TS }}\right) f_{||\mathbf Q _{R,A}^{\tau \mathrm TS }||_1}(y)\,\mathrm d y \\&\quad =1-\int \limits _0^{+\infty }\exp \left(-\frac{\gamma _\mathrm{th ,A} (\alpha ^{\tau \mathrm TS }+y)}{\beta ^{\tau \mathrm TS }}\right) f_{||\mathbf Q _{R,A}^{\tau \mathrm TS }||_1}(y)\,\text{ d}y \end{aligned}$$

where $\alpha ^{\tau \mathrm TS },\beta ^{\tau \mathrm TS }$ are notated as (25) and (26). With manipulation similar to the calculation from (46) to (47), the final result is derived as (24).

Appendix C: Proof of Theorem 2

The original integration of (31) is mathematically untractable when utilizing the exact expression of (23), (24) and (32), and we want to make use of the asymptotic CDF to calculate the integration. Following lemma is useful for asymptotic CDF derivation.

Lemma 3

Suppose the CDF of $X_1$ has an asymptotic form of $F_{X_1}(x;\rho )=\sum _{n=1}^{+\infty }a_n\left(\frac{x}{\rho }\right)^n$ as $\rho \rightarrow +\infty$, and CDF of $X_2$ $F_{X_2}(x;\rho )=\sum _{n=1}^{+\infty }b_n\left(\frac{x}{\rho }\right)^n$, then the CDF of a sum RV $Y=X_1+X_2$ has the asymptotic form CDF:

$$\begin{aligned} F_Y(y)=\int \limits _0^{y}\sum _{m=1}^{+\infty }a_m\left(\frac{y-x}{\rho } \right)^m\times \sum _{n=1}^{+\infty }\frac{nb_n}{\rho }\left(\frac{x}{\rho }\right) ^{n-1}\text{ d}x =\frac{a_1b_1y^2}{2\rho ^2}+o\left(\rho ^{-2}\right) \end{aligned}$$

(48)

$\square$

An example of Lemma 3 utilization can be seen in [30]. Using (29), (30) and the asymptotic form of (32)

$$\begin{aligned} F_{\gamma _{A,B}}(x)\approx \frac{x}{\bar{\rho }_{A,B}} \varTheta _1(\sigma _n^2,\bar{\mathbf{Q }}_A^{2\mathrm TS }) \end{aligned}$$

(49)

along with Lemma 3, Theorem 2 is immediately obtained.