Outage Probability and Ergodic Capacity Analysis for Two-Way Relaying System with Different Relay Selection Protocols

Abstract

The outage probability and ergodic capacity analysis for decode-and-forward two-way relaying system is investigated in this paper. First, the exact expressions (or bounds) of outage probability, ergodic capacity and average bidirectional ergodic capacity (ABEC) for max–min relay selection, random relay selection and direct transmission protocols are derived through theoretic analysis, and performance comparisons among different relay selection protocols are developed. Then a novel maximum average bidirectional mutual information (MABM) relay selection protocol is proposed and analyzed. Simulation results demonstrate that the derived analytical results fit well with Monte-Carlo simulations. The proposed MABM protocol can always achieve larger ABEC than other protocols while keeping low outage probability, and the MABM and max–min protocols in this paper can always achieve better performance than the max–min selection and max-sum selection in Krikidis (IEEE Trans Veh Technol 59(9):4620–4628, 2010). In addition, outage probability, ergodic capacity and ABEC performance of the proposed protocol become worse while distance becomes larger.

Appendix

1.1 Analysis of (11)

Let $z_1 = | {h_{12}} |^{2}$, thus $z_1$ is an exponentially distributed variable with parameter $\lambda _z =(d_1 +d_2)^{2}$, so

$$\begin{aligned} \mathcal{C }_{12\_dire}&= \int \limits _0^{+\infty } {\log _2} (1+\rho z_1 )\lambda _z e^{-\lambda _z z_1}dz_1 =\int \limits _1^{+\infty } {\frac{1}{\rho }} \log _2 (t)\lambda _z e^{-\lambda _z (t-1)/\rho }dt \nonumber \\&= -\log _2 (e)\cdot e^{\lambda _z /\rho }\mathbf{Ei}\left( {-\frac{\lambda _z}{\rho }} \right) \end{aligned}$$

(24)

where $\mathbf{Ei} (\cdot )$ denotes the exponential-integral function and the third equality follows from Eq. 4.331.2 in [27]. Recall that the exponential-integral function can have the following series representation from Eq. 8.214.1 in [27]

$$\begin{aligned} \mathbf{Ei}\left( x \right) =\mathbb{C }+\ln (-x)+\sum _{k=1}^\infty {\frac{x^{k}}{k\cdot k!}} ,\quad x<0 \end{aligned}$$

(25)

where $\mathbb{C }$ is Euler’s constant. Thus for high SNR, we can get Ei $(x )\approx \mathbb{C }+\ln (-x),x\rightarrow -0$. For larger SNR, we can have the following approximation

$$\begin{aligned} \mathcal{C }_{12\_dire} \approx -\log _2 (e)\cdot e^{\lambda _z /\rho }\left[ {\mathbb{C }+\ln \left( \frac{\lambda _z}{\rho }\right) } \right] \approx -\log _2 (e)\cdot \left[ {\mathbb{C }+\ln \left( \frac{\lambda _z}{\rho }\right) } \right] \end{aligned}$$

(26)

1.2 Analysis of (12)

Let $z_1 =| {h_{12}} |^{2}, z_2 = | {h_{21}} |^{2}$, thus $z_1$ and $z_2$ are independent exponentially distributed variable and both with parameter $\lambda _z =(d_1 +d_2 )^{2}$, so

$$\begin{aligned} \mathcal{C }_{ave\_dire}&= \int \limits _0^{+\infty } {\int \limits _0^{+\infty } {\frac{1}{2}}} \log \left[ {\left( {1+\rho z_1} \right) \left( {1+\rho z_2} \right) } \right] \lambda _z e^{-\lambda _z z_1}\lambda _z e^{-\lambda _z z_2}dz_1 dz_2 \nonumber \\&= \frac{1}{2}\int \limits _0^{+\infty } {\int \limits _0^{+\infty } {\log \left[ {\left( {1+\rho z_1} \right) } \right] }} \times \lambda _z e^{-\lambda _z z_1}\lambda _z e^{-\lambda _z z_2}dz_2 dz_1 \nonumber \\&\quad +\frac{1}{2}\int \limits _0^{+\infty } {\int \limits _0^{+\infty } {\log \left[ {\left( {1+\rho z_2} \right) } \right] }} \times \lambda _z e^{-\lambda _z z_1}\lambda _z e^{-\lambda _z z_2}dz_1 dz_2 \nonumber \\&\approx -\log _2(e)\cdot \left[ {\mathbb{C }+\ln (\frac{\lambda _z}{\rho })} \right] \end{aligned}$$

(27)

1.3 Analysis of (14)

Let $x= | {h_{1R^{*}}} |^{2},\, y= | {h_{2R^{*}}} |^{2}$, $\varepsilon ={(2^{r}-1)}/\rho$, thus $x$ and $y$ are exponentially distributed variables with parameters $\lambda _x =d_1^2$ and $\lambda _y =d_2^2$, respectively, so

$$\begin{aligned} \mathcal{C }_{12\_rand}&= \int \limits _\varepsilon ^{+\infty } {\log (1+\rho y)} \lambda _y e^{-\lambda _y y}dy\nonumber \\&= \frac{\log _2 (e)}{\rho }\lambda _y e^{{\lambda _y}/\rho }2^{r}\int \limits _1^{+\infty } {(\ln 2^{r}+\ln t)} e^{{-\lambda _y 2^{r}t}/\rho }dt \end{aligned}$$

(28)

Similar to analysis of (11), we can get

$$\begin{aligned} \mathcal{C }_{12\_rand}&\approx \log _2 (e)\cdot e^{{\lambda _y }/\rho } \left[ \ln 2^{r}e^{{-\lambda _y 2^{r}}/\rho }-\mathbb{C }-\ln \left( \frac{\lambda _y 2^{r}}{\rho }\right) \right] \nonumber \\&\approx -\log _2 (e)\cdot \left[ {\mathbb{C }+\ln \left( \frac{\lambda _y}{\rho }\right) } \right] \end{aligned}$$

(29)

where the lower limit of integral is $\varepsilon$ because of the successful decoding condition $| {h_{R^{*}2}} |^{2}= | {h_{2R^{*}}} |^{2}\ge \varepsilon$ mentioned before.

1.4 Analysis of (18)

Let $x=| {h_{1R_i}} |^{2},y=| {h_{2R_i}} |^{2}, \xi =\mathop {\max }\nolimits _{R_i \in F} \min [ {| {h_{1R_i}} |^{2},| {h_{2R_i}} |^{2}} ]$, thus the PDF of $\xi$ can be shown as [28] $f_\xi (\xi )=N(\lambda _x+\lambda _y )e^{-(\lambda _x+\lambda _y)\xi }[{1-e^{-(\lambda _x+\lambda _y)\xi }} ]^{N-1}$, where $\lambda _x =d_1^2$ and $\lambda _y =d_2^2$ represent distributed parameters of exponentially distributed variables $| {h_{1R_i}} |^{2}$ and $| {h_{2R_i}} |^{2}$, respectively.

If we choose $\mathop {\max }\nolimits _{R_i \in F} \min [ { | {h_{1R_i}} |^{2}, | {h_{2R_i}} |^{2}} ]$ as $| {h_{2R^{*}}} |^{2}$ in the calculation of ergodic capacity every time when we choose $R^{*}$, i.e., let $| {h_{2R^{*}}} |^{2}=\xi$, we can get the lower bound,

$$\begin{aligned} \mathcal{C }_{12\_mm}&\ge \mathcal{C }_{12\_mm\_down}= \sum _{N=0}^L {\left\{ {\mathcal{C }_{mm1}} \right\} \cdot P\left( {\mathbf{N}=N} \right) }\end{aligned}$$

(30)

$$\begin{aligned} \mathcal{C }_{mm1}&= \int \limits _\varepsilon ^{+\infty } {\log (1+\rho \xi )} f_\xi (\xi )d\xi \nonumber \\&= \int \limits _{2^{r}}^{+\infty } {\log t} \frac{1}{\rho }N(\lambda _x +\lambda _y)e^{-(\lambda _x +\lambda _y)(t-1)/\rho }\left[ {1-e^{-(\lambda _x +\lambda _y)(t-1)/\rho }} \right] ^{N-1}dt\qquad \end{aligned}$$

(31)

Applying binomial expansion, we can have

$$\begin{aligned} \mathcal{C }_{mm1}&= \int \limits _{2^{r}}^{+\infty } {\log t} \frac{1}{\rho }N(\lambda _x +\lambda _y)e^{-(\lambda _x+\lambda _y )(t-1)/\rho }\left[ {\sum _{i=0}^{N-1} {C_{N-1}^i} (-1)^{i}e^{-(\lambda _x+\lambda _y)(t-1)i/\rho }} \right] dt \nonumber \\&\approx \frac{N\log _2 (e)}{\rho }(\lambda _x +\lambda _y)\sum _{i=0}^{N-1} {C_{N-1}^i} (-1)^{i}\int \limits _{2^{r}}^{+\infty } {e^{-(\lambda _x +\lambda _y)(1+i)t/\rho }} \ln tdt \end{aligned}$$

(32)

For

$$\begin{aligned} \int \limits _{2^{r}}^{+\infty } {e^{-(\lambda _x +\lambda _y )(1+i)t/\rho }} \ln tdt&= 2^{r}\int \limits _1^{+\infty } {e^{-(\lambda _x +\lambda _y)(1+i)2^{r}t/\rho }} (\ln t+\ln 2^{r})dt \nonumber \\&= 2^{r}\left( {-\frac{1}{\mu }} \right) \mathbf{Ei}\left( {-\mu } \right) +2^{r}\ln 2^{r}\frac{1}{\mu }e^{-\mu } \end{aligned}$$

(33)

where $\mu =(\lambda _x +\lambda _y)2^{r}(1+i)/\rho$, combining (32) and (33),

$$\begin{aligned} \mathcal{C }_{mm1}&= N\log _2 (e)\sum _{i=0}^{N-1} {C_{N-1}^i} (-1)^{i+1}\frac{1}{i+1}\left[ {\mathbf{Ei}\left( {-\mu } \right) -\ln 2^{r}e^{-\mu }} \right] \nonumber \\&\approx N\log _2 (e)\sum _{i=0}^{N-1} {C_{N-1}^i} (-1)^{i+1}\frac{1}{i+1}\nonumber \\&\times \left[ \mathbb{C }+\ln \left( {\frac{(\lambda _x +\lambda _y)2^{r}(1+i)}{\rho }} \right) -\ln 2^{r}e^{-\frac{(\lambda _x +\lambda _y)2^{r}(1+i)}{\rho }}\right] =-\log _2 (e)\nonumber \\&\times \left[ {\mathbb{C }+\ln (\frac{\lambda _x +\lambda _y }{\rho })} \right] +N\cdot \log _2 (e)\cdot \sum _{i=0}^{N-1} {C_{N-1}^i} (-1)^{i+1}\frac{1}{i+1}\ln (1+i)\qquad \end{aligned}$$

(34)

The last equality follows from Eq. 0.155.1 in [27].

1.5 Analysis of (19)

Let $x= | {h_{1R_i}} |^{2}, y= | {h_{2R_i}} |^{2}, \eta =\mathop {\max }\nolimits _{R_i \in F} \max [ { | {h_{1R_i}} |^{2}, | {h_{2R_i}} |^{2}} ]$, thus the PDF of $\eta$ can be shown as [28] $f_\eta (\eta )=N ({\lambda _x e^{-\lambda _x \eta }+\lambda _y e^{-\lambda _y \eta }-(\lambda _x +\lambda _y )e^{-(\lambda _x+\lambda _y)\eta }} ) [ (1-e^{-\lambda _x \eta }) (1-e^{-\lambda _y \eta }) ]^{N-1}$, where $\lambda _x =d_1^2$ and $\lambda _y =d_2^2$ represent distributed parameters of exponentially distributed variables $| {h_{1R_i}} |^{2}$ and $| {h_{2R_i}} |^{2}$, respectively.

If we choose $\mathop {\max }\nolimits _{R_i \in F} \max [ { | {h_{1R_i}} |^{2}, | {h_{2R_i}} |^{2}} ]$ as $| {h_{2R^{*}}} |^{2}$ in the calculation of ergodic capacity every time when we choose $R^{*}$, i.e., let $| {h_{2R^{*}}} |^{2}=\eta$, we can get the upper bound,

$$\begin{aligned} \mathcal{C }_{12\_mm} \le \mathcal{C }_{12\_mm\_up1} =\sum _{N=0}^L {\left\{ {\mathcal{C }_{mm}^*} \right\} P\left( {\mathbf{N}=N} \right) } \end{aligned}$$

(35)

When $\lambda _x <\lambda _y$,

$$\begin{aligned} \mathcal{C }_{mm}^*&= \int \limits _\varepsilon ^{+\infty } {\log (1+\rho \eta )} f_\eta (\eta )d\eta \nonumber \\&\le \int \limits _\varepsilon ^{+\infty } {\log } (1+\rho \eta )N(\lambda _x e^{-\lambda _x \eta }+\lambda _y e^{-\lambda _y \eta }-(\lambda _x +\lambda _y)e^{-(\lambda _x+\lambda _y)\eta }) \nonumber \\&\left[ {(1-e^{-\lambda _y \eta })} \right] ^{2N-2}d\eta = \int \limits _\varepsilon ^{+\infty } N \log (1+\rho \eta )\lambda _x e^{-\lambda _x \eta }\left[ {(1-e^{-\lambda _y \eta })} \right] ^{2N-2}d\eta \nonumber \\&\quad +\int \limits _\varepsilon ^{+\infty } N \log (1+\rho \eta )\lambda _y e^{-\lambda _y \eta }\left[ {(1-e^{-\lambda _y \eta })} \right] ^{2N-2}d\eta \nonumber \\&\quad -\int \limits _\varepsilon ^{+\infty } N \log (1+\rho \eta )(\lambda _x +\lambda _y)e^{-(\lambda _x+\lambda _y)\eta }\left[ {(1-e^{-\lambda _y \eta })} \right] ^{2N-2}d\eta \nonumber \\&= \mathcal{C }_{mm2} +\mathcal{C }_{mm3} +\mathcal{C }_{mm4} \end{aligned}$$

(36)

Similar to the analysis of (18), we can have

$$\begin{aligned}&\mathcal{C }_{mm2} \approx N\cdot \log _2 (e)\cdot \lambda _x \sum _{i=0}^{2N-2} {C_{2N-2}^i} (-1)^{i}\frac{1}{\lambda _x +\lambda _y i}\left[ {-\mathbb{C }-\ln \left( \frac{\lambda _x}{\rho }+\frac{\lambda _y}{\rho }i\right) } \right] \qquad \end{aligned}$$

(37)

$$\begin{aligned}&\mathcal{C }_{mm3} \approx N\cdot \log _2 (e)\cdot \sum _{i=0}^{2N-2} {C_{2N-2}^i} (-1)^{i}\frac{1}{1+i}\left[ {-\mathbb{C }-\ln \left( \frac{\lambda _y}{\rho }+\frac{\lambda _y}{\rho }i\right) } \right] \end{aligned}$$

(38)

$$\begin{aligned}&\mathcal{C }_{mm4} \approx N\cdot \log _2 (e)\cdot (\lambda _x +\lambda _y)\sum _{i=0}^{2N-2} {C_{2N-2}^i} (-1)^{i}\frac{1}{\lambda _x +\lambda _y +\lambda _y i}\nonumber \\&\quad \times \left[ {\mathbb{C }+\ln (\frac{\lambda _x +\lambda _y}{\rho }+\frac{\lambda _y}{\rho }i)} \right] \end{aligned}$$

(39)

1.6 Analysis of (20)

Like (19), we can get (40)–(42) for $\lambda _x \ge \lambda _y$,

$$\begin{aligned} \mathcal{C }_{mm5}&\approx N\cdot \log _2 (e)\cdot \sum _{i=0}^{2N-2} {C_{2N-2}^i} (-1)^{i}\frac{1}{1+i}\left[ {-\mathbb{C }-\ln (\frac{\lambda _x}{\rho }+\frac{\lambda _x }{\rho }i)} \right] \end{aligned}$$

(40)

$$\begin{aligned} \mathcal{C }_{mm6}&\approx N\cdot \log _2 (e)\cdot \lambda _y \sum _{i=0}^{2N-2} {C_{2N-2}^i} (-1)^{i}\frac{1}{\lambda _y +\lambda _x i}\left[ {-\mathbb{C }-\ln (\frac{\lambda _y}{\rho }+\frac{\lambda _x}{\rho }i)} \right] \end{aligned}$$

(41)

$$\begin{aligned} \mathcal{C }_{mm7}&\approx N\cdot \log _2 (e)\cdot (\lambda _x +\lambda _y)\sum _{i=0}^{2N-2} {C_{2N-2}^i} (-1)^{i}\frac{1}{\lambda _x +\lambda _y +\lambda _x i}\nonumber \\&\times \left[ {\mathbb{C }+\ln (\frac{\lambda _x +\lambda _y}{\rho }+\frac{\lambda _x}{\rho }i)} \right] \end{aligned}$$

(42)

1.7 Analysis of (23)

Let $x_1 =\rho | {h_{1R_i}} |^{2},\,y_1 =\rho | {h_{2R_i}} |^{2},\,z=(1+x_1)(1+y_1)$, thus $x_1$ and $y_1$ are exponentially distributed variables with parameters $\lambda _1 =\frac{d_1^2}{\rho },\, \lambda _2 =\frac{d_2^2}{\rho }$, respectively, so the CDF of $z$ can be shown as

$$\begin{aligned} P(\mathrm{z}<z)&= ~~~~~~\int \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\int \limits _{(1+x_1)(1+y_1)<z} {\lambda _1} e^{-\lambda _1 x_1}\lambda _2 e^{-\lambda _2 y_1}dx_1 dy_1 \nonumber \\&= \int \limits _0^{z-1} {\int \limits _0^{\frac{z-1-y_1}{1+y_1}} {\lambda _1}} e^{-\lambda _1 x_1 }\lambda _2 e^{-\lambda _2 y_1}dx_1 dy_1 \nonumber \\&= 1-e^{-\lambda _2 (z-1)}-\lambda _2 e^{\lambda _1 +\lambda _2 }\int \limits _1^z {e^{-\frac{\lambda _1 z}{t}-\lambda _2 t}} dt \end{aligned}$$

(43)

Let $f(z)=1-e^{-\lambda _2 (z-1)}-\lambda _2 e^{\lambda _1 +\lambda _2}\int \nolimits _1^z {e^{-\frac{\lambda _1 z}{t}-\lambda _2 t}} dt$, $g(z)=1-e^{-\lambda _1 \lambda _2 z}$. Although currently we are not able to find the formal proof for this property, Fig. 11 shows that $f(z)\approx g(z)$ for high SNR. Thus we can regard $z$ as an exponentially distributed variable with parameter $\lambda _1 \lambda _2$ approximately.

$$\begin{aligned} P(\mathrm{z}<z)\approx 1-e^{-\lambda _1 \lambda _2 z} \end{aligned}$$

(44)

So we can have

$$\begin{aligned} \mathcal{C }_{ave\_mabm}&\approx \sum _{N=0}^L {\left\{ {\mathcal{C }_{mabm}} \right\} \cdot P\left( {\mathbf{N}=N} \right) }\end{aligned}$$

(45)

$$\begin{aligned} \mathcal{C }_{mabm}&= \int \limits _{2^{2r}}^{+\infty } {\frac{1}{2}} \log z\cdot N\lambda _1 \lambda _2 e^{-\lambda _1 \lambda _2 z}\left( {1-e^{-\lambda _1 \lambda _2 z}} \right) ^{N-1}dz \end{aligned}$$

(46)

Like analysis of (18), we can get

$$\begin{aligned} \mathcal{C }_{mabm} \approx \frac{N}{2}\log _2 (e)\cdot \sum _{i=0}^{N-1} {C_{N-1}^i (-1)^{i}} \frac{1}{1+i}\left\{ {-\mathbb{C }-\ln \left[ {\lambda _1 \lambda _2 (1+i)} \right] } \right\} \end{aligned}$$

(47)

Fig. 11

Comparison between $f(z)$ and $g(z)$