Journal of Electrical and Computer Engineering

Volume 2016 (2016), Article ID 9689586, 9 pages

http://dx.doi.org/10.1155/2016/9689586

## First-Order Statistical Characteristics of Macrodiversity System with Three Microdiversity MRC Receivers in the Presence of - Short-Term Fading and Gamma Long-Term Fading

^{1}Faculty of Technical Sciences, University of Pristina, Kneza Milosa 7, 38220 Kosovska Mitrovica, Serbia^{2}Faculty of Electrical Engineering, University of Nis, Aleksandra Medvedeva 14, 18000 Nis, Serbia^{3}Teachers College, University of Pristina, Nemanjina, 38218 Leposavic, Serbia

Received 25 July 2016; Accepted 1 November 2016

Academic Editor: George S. Tombras

Copyright © 2016 Branimir Jaksic 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

Macrodiversity system with macrodiversity SC receiver and three microdiversity MRC (maximum ratio combining) receivers is considered. Independent* k*-*μ* short-term fading and correlated Gamma long-term fading are present at the inputs of microdiversity MRC receivers. For this model, the probability density function and the cumulative density function of microdiversity MRC receivers and macrodiversity SC receiver output signal envelopes are calculated. Influences of Gamma shadowing severity,* k*-*μ* multipath fading severity, Rician factor and correlation coefficient at probability density function, and cumulative density function of macrodiversity SC receiver output signal envelopes are graphically presented.

#### 1. Introduction

Fading is a basic type of nuisance in wireless mobile telecommunication systems. Depending on propagation environment and different communications cases various types of fading can arise. Short-term fading is a result of signal propagation on multipath. The interaction of waves between transmitter and receiver (reflection, diffraction, and scattering) induces large numbers of sent copies signals on the input of receivers. Propagation environment can be linear and nonlinear. Nonlinear environment is defined as correlated surfaces in which dissipation field is not equal [1–3].

Long term fading arises because of shadow effect. Various objects create shadow effect in areas between transmitter and receiver. In most cases, long term fading is correlated. Changing of signal power due to the influence of shadow effect is slow in comparison to the signal envelope changing into short term fading. The signal envelope is variable due to short term fading, and the signal envelope power is variable due to the long term fading [1, 4].

The signal from the transmitter to the receiver can be propagated over one, two, or more clusters. Cluster is defined as waves which arrive at the inputs of receivers with approximately same delay. When the number of clusters increases, the fading severity decreases. Each cluster is formed by a pair of Gaussian components at the receiver [2, 5, 6].

The statistical behavior of signal in such systems can be described by different distributions as Rayleigh, Rice, Nakagami-*m*, Weibull, or* k*-*μ*.* k*-*μ* distribution can be used to describe the variation of the signal envelope in linear environments, with dominant component, several clusters in propagation environment, and equal components in quadrature of signal.* k*-*μ* distribution has two parameters. The parameter is Rician factor. Rician factor is defined as ratio of dominant components power and scattering components power. System performance is better for higher values of Rician factor. Rician factor increases as dominant components power increases or scattering components power decreases. The parameter *μ* is related to the number of clusters in propagation channels. The* k*-*μ* distribution is general distribution and several another distributions (Rician, Nakagami-*m,* and Rayleigh) can be obtained from -*μ* distribution as special cases [7, 8].

Various diversity techniques to reduce the impact of short-term fading and long-term fading on the system performance can be used. The most commonly used are spatial diversion techniques. Spatial diversity techniques are implemented with multiple antennas mounted on the receiver. By using spatial diversity technique the reliability of the system and the channel capacity increase without increasing the transmitter power and frequency band expansion. There are several spatial diversity combining techniques that can be used to reduce the influence of fading and cochannel interference on system performances. The most commonly used diversity techniques are MRC (maximum ratio combining), EGC (equal gain combining), and SC (selection combining) [2, 7].

MRC diversity technique gives the best results. This technique effectively reduces the influence of -*μ* short term fading on the system performance and provides the greatest diversity gain. The ratio of signal power and noise at the output of the MRC receiver is equal to the sum of the ratio signal power and noise at its inputs. If noise power is the same in all diversity branches, then squared output signal is equal to the sum of the squares of the signal at its input. This method requires that the signals at the inputs are brought to the same phase. Because of that this method of combining is very complex and expensive for real implementation [8, 9].

There are more works in open technical literature considering second order statistics of diversity systems. In [10], average crossing rate and average fade duration of macrodiversity SC receiver with two microdiversity maximum ratio combining (MRC) receivers operating over Gamma shadowed Nakagami-*m* multipath fading are calculated. Macrodiversity SC receiver selects microdiversity receiver with higher input power to serve to user. In [11], average fade duration and level crossing rate of macrodiversity SC receiver with two microdiversity MRC receivers operating over Rician fading multipath environment are derived.

#### 2. System Model

In this paper, macrodiversity system with macrodiversity SC (selection combining) receiver and three microdiversity MRC (maximum ratio combining) receivers is analyzed. At the inputs of microdiversity MRC receivers are independent* k- μ* short term fading and Gamma long term fading. Gamma long term fading is correlated. The correlation coefficient decreases with increasing distance between the antennas.

Microdiversity MRC receiver reduces -*μ* short term fading effects and macrodiversity SC receivers reduce Gamma long term fading effects on system performances. Obtained macrosystem is predicted for one cell in the cellular mobile radio systems. Microdiversity receivers are placed on the base stations for mobile users of this cell. Macrodiversity systems used signals from several base stations placed in one cell (two or more) [2].

The system that is being considered is shown in Figure 1. The signals at the inputs and the outputs of the MRC receivers are denoted as in Figure 1. The signal at the output of macrodiversity system is denoted with .