Performance analysis of semi-blind two-way AF relaying over generalized-k fading channels

Abstract

In this paper, we present analytical expressions for the outage probability, average symbol error rate (SER), and average sum-rate of two-way dual-hop opportunistic amplify-and-forward (AF) relaying over independent and not necessarily identical generalized-k (KG) fading channels. New closed-form expression for the gain of non-regenerative semi-blind relays is derived. This expression is used to obtain the equivalent signal-to-noise ratio (SNR). Based on the above mentioned formulas, tight bounds of some performance metrics, the outage probability, average SER, and average sum-rate, are derived for the semi-blind and channel state information (CSI) relaying. In order to prove the exactness of the proposed mathematical analysis, a selection of numerical results is provided.

Appendices

Appendix A

The upper bound for the average sum-rate of semi-blind opportunistic AF relaying over generalized-k fading channels can be written as [18]

$$ E\{R_{SB}\}=\frac{1}{2ln(2)} {\int}_{0}^{\infty} ln\left( \frac{\sqrt{P}~P_{R_{i}}(C-1) \sqrt{C-1} (t)^{\frac{3}{2}}}{4C}\right) f_{\gamma}(t) dt $$

(50)

where f _γ (t) is the PDF of γ given as $\gamma =max\left (\gamma _{1},\ldots ,\gamma _{L}\right )$. By doing some manipulations, Eq. 50 can be rewritten as

$$\begin{array}{@{}rcl@{}} E\{R_{SB} \}&=& \frac{L}{2^{L(1+\beta)-1 }ln(2) } \frac{1}{\left( \varGamma(m) \varGamma(k) \right)^{L}} {\int}_{0}^{\infty} ln\\ &&\qquad\times\left( \frac{\sqrt{P}~P_{R_{i}}(C-1)\sqrt{C-1} ~x^{3} }{32 ~C ~b~ \sqrt{b}}\right)\\ && K_{\alpha}(x) x^{L(1\,+\,\beta)\,-\,1 }\left(G^{2,1}_{1,3}\left(\begin{array}{cc}\left.\begin{array}{c}\overset{\frac{1-\beta}{2} }{\frac{\alpha}{2}, \frac{-\alpha}{2},\! \!-\frac{1+\beta}{2}} \end{array}~\right|&{\frac{x^{2}}{4}} \end{array}\right)\right)^{L-1} dx\\ \end{array} $$

(51)

The above equation can be written in the form ${\int }_{0}^{\infty } \exp (-x)g(x)dx$ by multiplying and dividing it with $\exp (x)$. Where $g(x)= \exp (x) f(x)$. Therefore, it can be solved using the Gauss-Laguerre quadrature as

$$\begin{array}{@{}rcl@{}} E\{R_{SB} \}&=& \frac{L}{2^{L(1+\beta)-1 }ln(2) }\frac{1}{\left( \varGamma(m) \varGamma(k) \right)^{L}} \sum\limits_{n=1}^{N } w_{n} \exp(c_{n}) ln\left( \frac{\sqrt{P}~P_{R_{i}}(C-1)\sqrt{C-1} ~{c_{n}^{3}} }{32 ~C ~b~ \sqrt{b}}\right)\\ && K_{\alpha}(c_{n}) c_{n}^{L(1+\beta)-1 }\left(G^{2,1}_{1,3}\left(\begin{array}{cc}\left.\begin{array}{c}\overset{\frac{1-\beta}{2} }{\frac{\alpha}{2}, \frac{-\alpha}{2}, -\frac{1+\beta}{2}} \end{array}~\right|&{\frac{c_{n}^{2}}{4}} \end{array}\right)\right)^{L-1} \end{array} $$

(52)

Appendix B

The upper bound for the average sum-rate of CSI-assisted opportunistic AF relaying over generalized-k fading channels can be written as [17]

$$ E\{R_{CSI} \}=\frac{1}{2ln(2)} {\int}_{0}^{\infty} ln\left(\frac{P_{R_{i}}^{2}}{\lambda_{i}}t^{2}\right) f_{\gamma}(t) dt $$

(53)

where f _γ (t) is the PDF of γ given as $\gamma =max\left (\gamma _{1},\ldots ,\gamma _{L}\right )$. After simple mathematical manipulations, the Eq. 53 can be rewritten as

$$\begin{array}{@{}rcl@{}} E\{R_{CSI}\}&=& \frac{L}{2^{L(1+\beta)-1 }ln(2)} \frac{1}{\left( \varGamma(m) \varGamma(k) \right)^{L}} {\int}_{0}^{\infty} ln\left( \frac{P_{R_{i}}^{2} x^{4}}{16~\lambda_{i}~b^{2}}\right)\\ && K_{\alpha}(x) x^{L(1+\beta)-1 }\left(G^{2,1}_{1,3}\left(\begin{array}{cc}\left.\begin{array}{c}\overset{\frac{1-\beta}{2}}{\frac{\alpha}{2}, \frac{-\alpha}{2}, -\frac{1+\beta}{2}} \end{array}~\right|&{\frac{x^{2}}{4}} \end{array}\right)\right)^{L-1}\!\! dx\\ \end{array} $$

(54)

which can be evaluated by the same method presented in Eq. 52 as

$$\begin{array}{@{}rcl@{}} E\{R_{CSI} \}&=& \frac{L}{2^{L(1+\beta)-1}ln(2)} \frac{1}{\left(\varGamma(m) \varGamma(k)\right)^{L}} \sum\limits_{n=1}^{N } w_{n} \exp(c_{n}) ln\left(\frac{P_{R_{i}}^{2} {c_{n}^{4}}}{16 ~b^{2}~\lambda_{i}}\right)\\ &&K_{\alpha}(c_{n}) c_{n}^{L(1+\beta)-1}\left(G^{2,1}_{1,3}\left(\begin{array}{cc}\left.\begin{array}{c}\overset{\frac{1-\beta}{2} }{\frac{\alpha}{2}, \frac{-\alpha}{2}, -\frac{1+\beta}{2}} \end{array}~\right|&{\frac{c_{n}^{2}}{4} } \end{array}\right)\right)^{L-1} \end{array} $$

(55)