Wireless Communications and Mobile Computing

Volume 2019, Article ID 1613690, 11 pages

https://doi.org/10.1155/2019/1613690

## Performance Analysis of AF Relays with Maximal Ratio Combining in Nakagami- Fading Environments

^{1}The School of Information Engineering, Nanchang University, Nanchang 330031, China^{2}The School of Information Engineering, East China Jiaotong University, Nanchang 330013, China

Correspondence should be addressed to Dong Qin; nc.ude.ues@gnodniq

Received 21 August 2018; Revised 17 December 2018; Accepted 10 February 2019; Published 24 February 2019

Academic Editor: Olivier Berder

Copyright © 2019 Dong Qin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates the maximal ratio combining (MRC) performance of an amplify and forward (AF) relay system in Nakagami- fading environments. The study considers a general scenario with distinct fading parameters for the following three links, source to relay link, and source to destination link and relay to destination link. We derive new closed form expressions for the statistics of important performance metrics, including the moment generating function, outage probability, higher order moments of equivalent signal to noise ratio (SNR), ergodic capacity, and average symbol error probability (SEP) of common modulation types. In particular, we focus on analytical SEP expressions in the context of an additive white generalized Gaussian noise (AWGGN). As an active area of research, generalized noise receives much attention for its flexible model. However, analytical performance of modulation scheme in generalized noise type has not been found in open literature for AF relaying with MRC despite its practical usefulness. Without the help of analytical solutions, the SEP in generalized noise can only be obtained by a large number of repeated simulation experiments. Therefore, we present the general SEP expression by using special Fox’s function. Simulation results verify the accuracy of our theoretical analysis and show that the diversity order of MRC criterion linearly depends upon Nakagami parameters of three links.

#### 1. Introduction

Maximal ratio combining (MRC) has been shown to achieve the effective reception diversity in traditional direct communications. In a relaying system, the destination can receive signals from direct source to destination link and relay link by using MRC [1–3]. A maximum diversity order was shown in [4] for a fixed gain amplify and forward (AF) relay when MRC is used at the destination. The outage performance of zero forcing beamforming and MRC for an AF two-way underlay network was analyzed in [5] while the outage probability and approximate symbol error rate of a multiple input multiple output relay with MRC was presented in [6, 7].

Although MRC has been extensively studied, most works have focused on deriving approximate solutions or upper and lower bound performance. Exact performance expressions are important in evaluating MRC but still lack sufficient study. In fact, the performance analysis of MRC encounters enormous computational complexity and inaccuracy, making it difficult to evaluate the comprehensive fading characteristics.

In this paper, we consider an AF relaying system with the MRC receiver in a Nakagami- fading environment, in which it is generally considered difficult to find the closed form expressions for performance study, especially when the links have distinct fading parameters. In contrast to the aforementioned works, the precise performance statistics of AF system with MRC receiver are analyzed in a Nakagami- fading environment. The relay system assumes that an indirect relay link and a direct link between source and destination coexist simultaneously. We first obtain the probability density function (PDF) of equivalent signal to noise ratio (SNR) according to the convolution formula. Then the cumulative distribution function (CDF) is presented in accordance with the principle of Laplace inverse transform. New exact formulas are developed for the statistics of higher order moments of equivalent SNR, ergodic capacity, and average SEP with distinct fading parameters among different hops. In addition, our SEP expression also permits assessment of error probability in the context of an additive white generalized Gaussian noise (AWGGN) environment, which notably facilitates more general characterization of noise types.

#### 2. System Model

A two-hop cooperative system with one source node S, one AF relay node R, and one destination node D is considered, as shown in Figure 1. The source node, relay node, and destination node are equipped with one antenna. Let , , and represent the channel coefficients for SR, RD, and SD links, respectively. Assume all the links undergo independent but not necessarily identically distributed Nakagami- fading. The instantaneous SNRs for direct SD link and indirect relay SRD link are, respectively, given bywhere and denote available transmit powers of the source node and the relay node, respectively. and are noise variances at the relay node and the destination node, respectively. The instantaneous SNR of the MRC combiner output is given by . The PDF of the gamma distributed variable is easily given by [7]where is the fading severity parameter of the direct link, denotes the corresponding scale parameters, and is the gamma function. When the fading parameter of both SR and RD links is a nonnegative integer plus one half, the PDF of can be found in [8] where is the fading severity parameter, denotes the corresponding scale parameters for , and is the floor function. Since the PDF of the sum of two random variables is the convolution of the PDFs of these two variables, the PDF of MRC SNR is thus expressed as where represents convolution operation, is beta function [9, eq.(8.380.1)], and is called the generalized hypergeometric function [9, eq.(9.14)]. Although the PDF of seems a little complicated, it only includes the finite multiplication and addition of exponential function, power function, and hypergeometric function, which can be easily calculated by the prevailing mathematical software, such as Maple.