Research on the Nth-best Relay Selection with Outdated Feedback in Selection Cooperation Systems

Abstract

In this paper, the process of the Nth-best relay selection strategy with outdated feedback is given in the selection cooperative system. The performance of the proposed Nth-best relay selection with outdated channel state information is investigated in Rayleigh fading scenarios. The exact closed-form expression for the outage performance is derived. In the high signal-to-noise ratio regime, the asymptotic outage behavior is also analyzed. In addition, based on the asymptotic analysis, the diversity-multiplexing tradeoff of the considered system is obtained. At last, simulation results show that even a small deviation of the channel estimates from the actual values will lead to a severe degradation of the outage performance of the considered system.

Appendices

Appendix 1

In the case of $0 \le \rho < 1$, satisfying ${\gamma _{{k^*}D}} \ne {\tilde{\gamma } _{{k^*}D}}$. Based on our problem formulation we can write the PDF ${f_{{\gamma _{{k^*}D}}}}\left( x \right)$ as follows

$$\begin{aligned} P\left\{ {{\gamma _{SD}} + {\gamma _{{k^*}D}} < {g_T}} \right\} = \int _0^{{g_T}} {{F_{{{\tilde{\gamma } }_{{k^*}D}}}}\left( {{g_T} - x} \right) {f_{{\gamma _{SD}}}}\left( x \right) } dx \end{aligned}$$

(24)

Substituting (8) and ${f_{{\gamma _{SD}}}}\left( x \right) = {{\exp ( - {x /{{{\bar{\gamma } }_{SD}}}})} /{{{\bar{\gamma } }_{SD}}}}$ into (24), after some algebraic manipulations, we can obtain

$$\begin{aligned} P\left\{ {{\gamma _{SD}} + {\gamma _{{k^*}D}} < {g_T}} \right\} = N\left( {\begin{array}{l} l\\ N \end{array}} \right) \sum \limits _{i = 0}^{l - N} {{{\left( { - 1} \right) }^i}\left( {\begin{array}{l} {l - N}\\ i \end{array}} \right) \frac{1}{{i + N}} \times } \left( {\int _0^{{g_T}} {\frac{1}{{{{\bar{\gamma } }_{SD}}}}{e^{ - \frac{x}{{{{\bar{\gamma } }_{SD}}}}}}} dx - {e^{ - \frac{{\left( {i + N} \right) {g_T}}}{{{{\bar{\gamma } }_{RD}}}}}}\int _0^{{g_T}} {\frac{1}{{{{\bar{\gamma } }_{SD}}}}{e^{ - \frac{{{{\bar{\gamma } }_{RD}} - \left( {i + N} \right) {{\bar{\gamma } }_{SD}}}}{{{{\bar{\gamma } }_{RD}}{{\bar{\gamma } }_{SD}}}}x}}} dx} \right) \end{aligned}$$

(25)

After some integral manipulations, (11) can be easily obtained from (25).

Appendix 2

In the case of $0 \le \rho < 1$, satisfying ${\gamma _{{k^*}D}} \ne {\tilde{\gamma } _{{k^*}D}}$. Based on our problem formulation we can write the PDF ${f_{{\gamma _{{k^*}D}}}}\left( x \right)$ as follow

$$\begin{aligned} {f_{{\gamma _{{k^*}D}}}}\left( x \right) = \int _0^\infty {{f_{{{{\gamma _{kD}}} / {{{\tilde{\gamma } }_{kD}} = z}}}}\left( x \right) {f_{{{\tilde{\gamma } }_{{k^*}D}}}}\left( z \right) } dy \end{aligned}$$

(26)

Using the Bayes rule, it is well known that the conditional probability density function can be expressed as

$$\begin{aligned} {f_{{{{\gamma _{kD}}} /{{{\tilde{\gamma } }_{kD}} = z}}}}\left( x \right) = \frac{{{f_{{\gamma _{kD}},{{\tilde{\gamma } }_{kD}}}}\left( {x,z} \right) }}{{{f_{{{\tilde{\gamma } }_{kD}}}}\left( z \right) }} \end{aligned}$$

(27)

Substituting (27) into (26), the PDF of ${\gamma _{{k^*}D}}$ can be rewritten as

$$\begin{aligned} {f_{{\gamma _{{k^*}D}}}}\left( x \right) = \int _0^\infty {\frac{{{f_{{\gamma _{kD}},{{\tilde{\gamma } }_{kD}}}}\left( {x,y} \right) }}{{{f_{{{\tilde{\gamma } }_{kD}}}}\left( y \right) }}{f_{{{\tilde{\gamma } }_{{k^*}D}}}}\left( y \right) } dy \end{aligned}$$

(28)

Substituting (4) and (9) into (28), after some simpler manipulations, we can obtain (12).

Appendix 3

In case of $0 \le \rho < 1$, the term $P\left\{ {{\gamma _{SD}} + {\gamma _{{k^*}D}} < {g_T}} \right\}$ in (10) can be written as

$$\begin{aligned} P\left\{ {{\gamma _{SD}} + {\gamma _{{k^*}D}} < {g_T}} \right\} = \int _0^{{g_T}} {{F_{{\gamma _{{k^*}D}}}}\left( {{g_T} - x} \right) {f_{{\gamma _{SD}}}}\left( x \right) } dx \end{aligned}$$

(29)

Substituting (13) and ${f_{{\gamma _{SD}}}}\left( x \right) = {{\exp ( - x /{{\bar{\gamma } }_{SD}}}}$ into (29), after some simple manipulations, we obtain

$$\begin{aligned} P\left\{ {{\gamma _{SD}} + {\gamma _{{k^*}D}} < {g_T}} \right\} = N\left( {\begin{array}{l} l\\ N \end{array}} \right) \sum \limits _{i = 0}^{l - N} {\left( {\begin{array}{l} {l - N}\\ i \end{array}} \right) \frac{{{{\left( { - 1} \right) }^i}}}{{i + N}} \times \left( {\int _0^{{g_T}} {\frac{1}{{{{\bar{\gamma } }_{SD}}}}{e^{ - \frac{x}{{{{\bar{\gamma } }_{SD}}}}}}} dx - } \right. } \left. {{e^{ - \frac{{\left( {i + N} \right) {g_T}}}{{\left[ {1 + \left( {i + N - 1} \right) \left( {1 - {\rho ^2}} \right) } \right] {{\bar{\gamma } }_{RD}}}}}}\int _0^{{g_T}} {\frac{1}{{{{\bar{\gamma } }_{SD}}}}{e^{ - \frac{{\left[ {1 + \left( {i + N - 1} \right) \left( {1 - {\rho ^2}} \right) } \right] {{\bar{\gamma } }_{RD}} - \left( {i + N} \right) {{\bar{\gamma } }_{SD}}}}{{\left[ {1 + \left( {i + N - 1} \right) \left( {1 - {\rho ^2}} \right) } \right] {{\bar{\gamma } }_{RD}}{{\bar{\gamma } }_{SD}}}}x}}} dx} \right) \end{aligned}$$

(30)

Then, (14) can be easily derived from (30).