Performance Analysis of MRC Technique in DF Cognitive Relay Networks

Abstract

A common combing technique that Maximum Ratio Combing (MRC) in decode-and-forward (DF) cognitive relay network has been studied in this paper. The direct links are available and the maximum relay selection is utilized to select the ideal relay for communication in our system, and all channels are Rayleigh fading channels. We present the secrecy outage probability (SOP) for measuring the system performance and observe that the diversity order is $K+1$ from the asymptotic SOP, where K refers to the number of relays. Simulation results are also provided to inspect the veracity of analytic results.

Appendix

Based on $Z = \frac{I}{{\left( {\rho - 1} \right) {N_{{0_D}}}}}{\left| {{h_D}} \right| ^2} - \frac{{I\rho }}{{\left( {\rho - 1} \right) {N_{{0_E}}}}}{\left| {{h_E}} \right| ^2}$, the CDF of Z is rewritten as

$$\begin{aligned}&{F_Z}\left( z \right) = \Pr \left( {\frac{I}{{\left( {\rho - 1} \right) {N_{{0_D}}}}}{{\left| {{h_D}} \right| }^2} - \frac{{I\rho }}{{\left( {\rho - 1} \right) {N_{{0_E}}}}}{{\left| {{h_E}} \right| }^2} < z} \right) \nonumber \\&= \int _0^\infty {{F_X}\left( {\frac{{\left( {\rho - 1} \right) {N_{{0_D}}}}}{I}z + \frac{{\rho {N_{{0_D}}}}}{{{N_{{0_E}}}}}y} \right) {f_Y}\left( y \right) dy,} \end{aligned}$$

(19)

where $X = {\left| {{h_D}} \right| ^2}$ and $Y = {\left| {{h_E}} \right| ^2}$, and we should obtain the CDF of X and the PDF of Y first. According to ${\left| {{h_D}} \right| ^2} = {{{\left| {{h_{SD}}} \right| }^2}+\mathop {\text {max}}\limits _{1 \le k \le K}\eta {{\left| {{h_{{R_k}D}}} \right| }^2}}$, the PDF of X is

$$\begin{aligned}&{F_X}\left( x \right) = \Pr \left( {{{\left| {{h_{SD}}} \right| }^2} + \mathop {\text {max}}\limits _{1kK} \eta {{\left| {{h_{{R_k}D}}} \right| }^2}< x} \right) \nonumber \\&= \Pr \left( {{{\left| {{h_{SD}}} \right| }^2} < x - \mathop {\text {max}}\limits _{1kK} \eta {{\left| {{h_{{R_k}D}}} \right| }^2}} \right) = \int _0^x {{F_{\mathop {\text {max}}\limits _{1kK} \eta {{\left| {{h_{{R_k}D}}} \right| }^2}}}} \left( {x - l} \right) {f_{{{\left| {{h_{SD}}} \right| }^2}}}\left( l \right) dl. \end{aligned}$$

(20)

According to the Rayleigh fading of R-D link, ${{F_{\mathop {\text {max}}\limits _{1kK} \eta {{\left| {{h_{{R_k}D}}} \right| }^2}}}}$ is shown as

$$\begin{aligned} {F_{\mathop {\text {max}}\limits _{1kK} \eta {{\left| {{h_{{R_k}D}}} \right| }^2}}} = {\left( {1 - {e^{ - \frac{x}{{\eta {\mu _1}}}}}} \right) ^K} = \sum \limits _{n = 0}^K {\left( \!\begin{aligned} K \\ n \\ \end{aligned} \right) {{\left( { - 1} \right) }^n}{e^{ - \frac{{nx}}{{\eta {\mu _1}}}}}}. \end{aligned}$$

(21)

Substituting Eq. (21) and ${f_{{{\left| {{h_{SD}}} \right| }^2}}}\left( l \right) = \frac{1}{{{\lambda _1}}}{e^{ - \frac{l}{{{\lambda _1}}}}}$ into Eq. (20), we can obtain ${F_X}\left( x \right)$ as

$$\begin{aligned} {F_X}\left( x \right) = \sum \limits _{n = 0}^K {\left( \begin{aligned} K \\ n \\ \end{aligned} \right) {{\left( { - 1} \right) }^n}\frac{{\frac{1}{{{\lambda _1}}}}}{{\frac{n}{{\eta {\mu _1}}} - \frac{1}{{{\lambda _1}}}}}} \left( {{e^{ - \frac{x}{{{\lambda _1}}}}} - {e^{ - \frac{{nx}}{{\eta {\mu _1}}}}}} \right) . \end{aligned}$$

(22)

The derivation of ${F_Y}\left( y \right)$ is similar to the derivation of ${F_X}\left( x \right)$, and we present the result of ${F_Y}\left( y \right)$ directly

$$\begin{aligned} {F_Y}\left( y \right) = 1 - {e^{ - \frac{y}{{\eta {\mu _2}}}}} - \frac{{\frac{1}{{\eta {\beta _2}}}}}{{\frac{1}{{{\lambda _2}}} - \frac{1}{{\eta {\beta _2}}}}}\left( {{e^{ - \frac{y}{{\eta {\mu _2}}}}} - {e^{ - \frac{y}{{{\lambda _2}}}}}} \right) . \end{aligned}$$

(23)

Thus, the PDF of ${F_Y}\left( y \right)$ is

$$\begin{aligned} {f_Y}\left( y \right) = \frac{{d{F_Y}\left( y \right) }}{{dy}} = \frac{{\frac{1}{{\eta {\lambda _2}{\beta _2}}}}}{{\frac{1}{{{\lambda _2}}} - \frac{1}{{\eta {\beta _2}}}}}\left( {{e^{ - \frac{y}{{\eta {\mu _2}}}}} - {e^{ - \frac{y}{{{\lambda _2}}}}}} \right) . \end{aligned}$$

(24)

Then, substituting Eqs. (22) and (24) into Eq. (19), the CDF of Z is presented as in Corollary 1.