Performance analysis of maximal ratio transmission with receiver antenna selection over correlated Nakagami-m fading channels

Abstract

In this paper, we develop a framework to analyze the performance analysis of multi-antenna system that employ maximal-ratio transmission with receive antenna selection (MRT/RAS) over correlated Nakagami-m fading channels. We derive the probability density function (pdf) of the post processing signal-to-noise ratio at the output of the decoder, when spatial fading correlation is assumed. Utilizing the pdf, a closed-form analytical expression is derived in terms of channel capacity, outage probability and symbol error rate (SER) of M-ary modulation schemes. It can be observed that the performance loss in terms of capacity and SER is negligible when the separation between adjacent antennas is as large as 0.5 for exponential correlation model. Numerical outcomes considering the spatial correlation represent the performance characteristics and analyze their impacts on channel capacity, outage probability and SER of MRT/RAS system.

Appendices

Appendix 1

$$ \begin{aligned} C & \le C_{Bound} \\ & = log_{2} \left[ {1 + \mathop \int \limits_{0}^{\infty } \gamma p_{\gamma } \left( \gamma \right)d\gamma } \right] \\ \end{aligned} $$

(31)

Inserting (12) into (31), the upper bound capacity can be written as

$$ \begin{aligned} C & = log_{2} \left[ {1 + \mathop \sum \limits_{\sigma = 1}^{N} \mathop \sum \limits_{\tau = 1}^{{m\mu_{\sigma } }} \mathop \sum \limits_{v = 0}^{{N_{r} - 1}} \mathop \sum \limits_{{\left\{ {\delta_{n,p,c} } \right\} \in {\mathbb{U}}_{v} }} \left( {\begin{array}{*{20}c} {N_{r} } \\ {\left\{ {\delta_{n,p,c} } \right\}} \\ \end{array} } \right)\frac{{\xi_{\sigma \tau } h\left( {\left\{ {\delta_{n,p,c} } \right\}} \right)}}{{\left( { - 1} \right)^{v} \Gamma \left( \tau \right)}}} \right. \\ & \quad \left. { \times \mathop \int \limits_{0}^{\infty } \frac{{\gamma^{{\left[ {\tau + f\left( {\left\{ {\delta_{n,p,c} } \right\}} \right) + 1} \right] - 1}} e^{{ - \gamma \left( {g\left( {\left\{ {\delta_{n,p,c} } \right\}} \right) + \frac{m}{{\bar{\gamma }\lambda_{v} }}} \right)}} }}{{\left( {\frac{{\bar{\gamma }\lambda_{v} }}{m}} \right)^{\tau } \Gamma \left( {N_{r} - v} \right)}}d\gamma } \right] \\ \end{aligned} $$

(32)

By utilizing the identity [19] $\int_{0}^{\infty } {x^{v - 1} } e^{ - \mu x} dx = \mu^{ - v} \Gamma \left( v \right)$, the upper bound capacity can be obtained in (19).

Appendix 2

Inserting (12) and (28) into (24)

$$ \begin{aligned} & P_{M\_PSK}^{MRT} \left( s \right) = \mathop \sum \limits_{z = 1}^{2} \mathop \sum \limits_{v = 1}^{{max\left( {M/4,1} \right)}} \mathop \sum \limits_{\sigma = 1}^{N} \mathop \sum \limits_{\tau = 1}^{{m\mu_{\sigma } }} \mathop \sum \limits_{v = 0}^{{N_{r} - 1}} \mathop \sum \limits_{{\left\{ {\delta_{n,p,c} } \right\} \in {\mathbb{U}}_{v} }} \left( {\begin{array}{*{20}c} {N_{r} } \\ {\left\{ {\delta_{n,p,c} } \right\}} \\ \end{array} } \right)\frac{{a_{z} \xi_{\sigma \tau } }}{{\left( { - 1} \right)^{v} \Gamma \left( \tau \right)}}\frac{{\left( {\frac{m}{{\bar{\gamma }\lambda_{v} }}} \right)^{\tau } h\left( {\left\{ {\delta_{n,p,c} } \right\}} \right)}}{{2max\left( {log_{2} M,2} \right)}} \\ & \quad \times \mathop \int \limits_{0}^{\infty } \gamma^{{\tau + f\left( {\left\{ {\delta_{n,p,c} } \right\}} \right) - 1}} e^{{ - \gamma \left( {g\left( {\left\{ {\delta_{n,p,c} } \right\}} \right) + \frac{m}{{\bar{\gamma }\lambda_{v} }} + b_{z} D_{v} } \right)}} d\gamma \\ \end{aligned} $$

(33)

Utilizing the identity [19] $\int_{0}^{\infty } {x^{v - 1} } e^{ - \mu x} dx = \frac{1}{{\mu^{v} }}\Gamma \left( v \right)$, we can achieve approximate closed-form expression of average SER of MRT/RAS with MPSK in (29).