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

Authors: Branimir Jaksic; Mihajlo Stefanovic; Danijela Aleksic; Dragan Radenkovic; Sinisa Minic
Journal: Journal of Electrical and Computer Engineering
 (2016)

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## Abstract

Macrodiversity system with macrodiversity SC receiver and three microdiversity MRC (maximum ratio combining) receivers is considered. Independentk-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.


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## Body

## 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, ork-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 k 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. Thek-μ distribution is general distribution and several another distributions (Rician, Nakagami-m, and Rayleigh) can be obtained from k-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 ofk-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 independentk-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 reducesk-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 Figure1. 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 w.Figure 1
Macrodiversity system with three microdiversity MRC receivers and one macrodiversity SC receiver.The square of the macrodiversity system output signal envelope is equal to sum of signal squares from its inputs. The macrodiversity SC receiver output signal envelope is equal to the microdiversity MRC receiver output signal envelope whose signal power at the input microdivesity MRC receiver is greater than the signal power at the input of the other two microdivesity MRC receivers.Squaredk-μ distribution is equal to sum of2μ squares of Gaussian random independent with same variance variables. In this way we can determine the probability density function ofk-μ distribution. With these probability density functions we can determine cumulative density function ofk-μ random variables, characteristic function of the random variablek-μ, and moments of thek-μ random variable.Probability density function of macrodiversity SC receiver output signal envelope is equal to the probability density function of microdiversity MRC receiver output signal envelope with highest signal power at this input.The cumulative density function is equal to integrating of probability density function. Thus can be calculated the cumulative density function of output signal envelope for the first, second, and third microdiversity MRC receiver. The cumulative density function of macrodiversity SC receiver output signal envelope can be determined by integrating the probability density function of macrodiversity SC receiver output signal envelope. By using the cumulative density function can be determined outage probability [3].


## 3. Probability Density Function of Macrodiversity SC Receiver Output Signal Envelope

Squaredk-μx1 signal is equal to(1)x12=x112+x122+⋯+x12μ2.Squaredk-μx2 signal is equal to(2)x22=x212+x222+⋯+x22μ2.The squared signalx is equal to(3)x2=x12+x22=x112+x122+⋯+x12μ2+x212+x222+⋯+x22μ2=xk.Random variablexk has a χ2 distribution:(4)pxkxk=12σ2xkλ2μ-1/2e-λ+xk/2σ2I2μ+1λ1/2xk1/2σ2, where In(·) is modified Bessel function of the first kind, order n and argument x, σ signal variance, μ the number of clusters for the signal, and λ the mean signal power.Relations betweenxk and x are(5)x2=xk,x=xk1/2.Probability density function of microdiversity MRC receiver output signal envelopex is(6)pxx=dxkdxpxkx2,where(7)dxkdx=2x. By substituting (7) and (4) in (6) the development of Bessel functions is obtained:(8)pxx=1σ12λ2μ-1∑i1=0∞λ1/22σ122i1+2μ-11i1!Γi1+2μx2i1+4μ-1e-λ+x2/2σ12.One takes(9)2σ12=Ω1.After substituting (9) in (8), probability density function of microdiversity MRC receiver output signal envelope becomes(10)pxx=2Ω1λ2μ-1∑i1=0∞λi1+μ-1/2Ω1-i1-μ+1/21i1!Γi1+2μx2i1+4μ-1e-λ+x2/Ω1,where Ω1 means signal power.In a similar way we can obtain the probability density function of microdiversity MRC receiver output signal envelope for second and third microdiversity MRC receiver:(11)pyy=2Ω2λ2μ-1∑i2=0∞λi2+μ-1/2Ω2-i2-μ+1/21i2!Γi2+2μy2i2+4μ-1e-λ+y2/Ω2,(12)pzz=2Ω3λ2μ-1∑i3=0∞λi3+μ-1/2Ω3-i3-μ+1/21i3!Γi3+2μz2i3+4μ-1e-λ+z2/Ω3.Signal envelopes powers at inputs in microdiversity receivers are correlated. Signal envelope powersΩ1, Ω2, and Ω3 follow Gamma distribution:(13)pΩ1Ω2Ω3Ω1Ω2Ω3=1Γc1-ρ2ρc-1Ω0c+2Ω1Ω3c-1/2e-Ω1+Ω3+Ω21+ρ/Ω01-ρIc-12ρΩ01-ρΩ1Ω2Ic-12ρΩ01-ρΩ2Ω3,where In(·) is modified Bessel function of the first kind, order n, and argument x; Ω0 is mean square of signal power variation, ρ correlation coefficient, and c Gamma shadowing severity.Probability density function of macrodiversity SC receiver output signal envelopew is(14)pww=∫0∞dΩ1∫0Ω1dΩ2∫0Ω1dΩ3pxwΩ1pΩ1Ω2Ω3Ω1Ω2Ω3+∫0∞dΩ2∫0Ω2dΩ1∫0Ω2dΩ3pywΩ2pΩ1Ω2Ω3Ω1Ω2Ω3+∫0∞dΩ3∫0Ω3dΩ1∫0Ω3dΩ2pzwΩ3pΩ1Ω2Ω3Ω1Ω2Ω3=I1+I2+I3=2I1+I2.Functionspx(w/Ω1), py(w/Ω2), and pz(w/Ω3) are given in (10), (11), and (12), respectively.IntegralI1 is equal to(15)I1=2λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i1=0∞1i1!Γi1+2μλi1+μ-1/2x2i1+4μ-1∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-11j2!Γj2+c∫0∞dΩ1Ω1-1-i1-μ+1/2+j1+c-1e-λ+x2/Ω1-Ω1/Ω01-ρ∫0Ω1dΩ2Ω2j1+j2+c-1e-Ω21+ρ/Ω01-ρ∫0Ω1dΩ3Ω3j2+c-1e-Ω3/Ω01-ρ.After using [12] for resolving the second and third integrals in (15), I1 becomes(16)I1=2λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i1=0∞1i1!Γi1+2μλi1+μ-1/2x2i1+4μ-1∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-11j2!Γj2+cΩ01-ρ1+ρj1+j2+cΩ01-ρj2+c∫0∞dΩ1Ω1-1-i1-μ+1/2+j1+c-1e-λ+x2/Ω1-Ω1/Ω01-ργj1+j2+c,1+ρΩ01-ρΩ1γj2+c,1Ω01-ρΩ1, where γ(·) represents the lower incomplete Gamma function. After the developing Gamma function,(17)γn,x=1nxne-xF111,n+1,x=1nxne-x∑i=0∞n!n+i!xi,I1 becomes(18)I1=2λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i1=0∞1i1!Γi1+2μλi1+μ-1/2x2i1+4μ-1∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-11j2!Γj2+cΩ01-ρ1+ρj1+j2+cΩ01-ρj2+c∫0∞dΩ1Ω1-1-i1-μ+1/2+j1+c-1e-λ+x2/Ω1-Ω1/Ω01-ρ1j1+j2+c1+ρΩ1Ω01-ρj1+j2+ce-1+ρΩ1/Ω01-ρ∑j3=0∞j1+j2+c!j1+j2+c+j3!1+ρΩ1Ω01-ρj31j2+cΩ1Ω01-ρj2+ce-Ω1/Ω01-ρ∑j4=0∞j2+c!j2+c+j4!Ω1Ω01-ρj4.By using [12] the following is obtained:(19)I1=2λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i1=0∞1i1!Γi1+2μλi1+μ-1/2x2i1+4μ-1∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-11j2!Γj2+cΩ01-ρ1+ρj1+j2+cΩ01-ρj2+c1j1+j2+c1+ρΩ01-ρj1+j2+c∑j3=0∞j1+j2+c!j1+j2+c+j3!1+ρΩ01-ρj31j2+c1Ω01-ρj2+c∑j4=0∞j2+c!j2+c+j4!1Ω01-ρj4λ+x2Ω01-ρ3+ρ-1/4-i1/2-μ/2+j1+j2+j3/2+j3/2+3c/2K-1/2-i1-μ+2j1+2j2+j3+j4+3c2λ+x23+ρΩ01-ρ,where Kn(x) is modified Bessel function of the second kind, order n, and argument x.IntegralI2 is equal to(20)I2=2λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i2=0∞1i2!Γi2+2μλi2+μ-1/2y2i2+4μ-1∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-11j2!Γj2+c∫0∞dΩ2Ω2-1/2-i2-μ+j1+c-1e-λ+y2/Ω2-Ω21+ρ/Ω01-ρ∫0Ω2dΩ1Ω1j1+c-1e-Ω1/Ω01-ρ∫0Ω2dΩ3Ω3j2+c-1e-Ω3/Ω01-ρ.After the using procedure for solvingI1, I2 becomes(21)I2=2λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i2=0∞1i2!Γi2+2μλi2+μ-1/2y2i2+4μ-1∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-1Ω01-ρj1+j2+2cj2!Γj2+c1j1+c1Ω01-ρj1+c∑j3=0∞j1+c!j1+c+j3!1Ω01-ρj31j2+c1Ω01-ρj2+c∑j4=0∞j2+c!j2+c+j4!1Ω01-ρj4λ+y2Ω01-ρ3+ρ-1/4-i2/2-μ/2+j1+j2/2+j3/2+j4/2+3c/2K-1/2-i2-μ+2j1+j2+j3+j4+3c2λ+y23+ρΩ01-ρ.After replacing the integralsI1 and I2 in (14), we obtain an expression for PDF of macrodiversity SC receiver output signal envelope:(22)p=8λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i1=0∞λi1+μ-1/2i1!Γi1+2μ∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-11j2!Γj2+cx2i1+4μ-1Ω01-ρ1+ρj2+c1j1+j2+c∑j3=0∞j1+j2+c!j1+j2+c+j3!1+ρΩ01-ρj31j2+c1Ω01-ρj2+c∑j4=0∞j2+c!j2+c+j4!1Ω01-ρj4λ+x2Ω01-ρ3+ρ-1/4-i1/2-μ/2+j1+j2+j3/2+j3/2+3c/2K-1/2-i1-μ+2j1+2j2+j3+j4+3c2λ+x23+ρΩ01-ρ+y2i1+4μ-1Ω01-ρj2+cj1+c∑j3=0∞j1+c!j1+c+j3!1Ω01-ρj2+j3+c1j2+c∑j4=0∞j2+c!j2+c+j4!1Ω01-ρj4λ+y2Ω01-ρ3+ρ-1/4-i1/2-μ/2+j1+j2/2+j3/2+j4/2+3c/2K-1/2-i1-μ+2j1+j2+j3+j4+3c2λ+y23+ρΩ01-ρ.


## 4. Cumulative Density Function of Macrodiversity SC Receiver Output Signal Envelope

Probability density function of microdiversity MRC receiver output signal envelopex is(23)Fxx=∫0xdtpxt=1Ω1λ2μ-1e-λ/Ω1∑i1=0∞λi1+μ-1/2Ω1-i1-μ+1/21i1!Γi1+2μΩ1i1+2μγi1+2μ,x2Ω1,where px(t) is probability density function given by (10).In a similar way we can obtain the cumulative density function of microdiversity MRC receiver output signal envelope for second and third microdiversity MRC receiver:(24)Fyy=1Ω2λ2μ-1e-λ/Ω2∑i2=0∞λi2+μ-1/2Ω2-i2-μ+1/21i2!Γi2+2μΩ2i2+2μγi2+2μ,y2Ω2,(25)Fzz=1Ω3λ2μ-1e-λ/Ω3∑i3=0∞λi3+μ-1/2Ω1-i3-μ+1/21i3!Γi3+2μΩ1i3+2μγi3+2μ,z2Ω3.Cumulative density function of macrodiversity SC receiver output signal envelopew is(26)Fww=∫0∞dΩ1∫0Ω1dΩ2∫0Ω1dΩ3FxwΩ1pΩ1Ω2Ω3Ω1Ω2Ω3+∫0∞dΩ2∫0Ω2dΩ1∫0Ω2dΩ3FywΩ2pΩ1Ω2Ω3Ω1Ω2Ω3+∫0∞dΩ3∫0Ω3dΩ1∫0Ω3dΩ2FzwΩ3pΩ1Ω2Ω3Ω1Ω2Ω3=I1+I2+I3=2I1+I2.FunctionsFx(w/Ω1), Fy(w/Ω2), and Fz(w/Ω3) are given by (23), (24), and (25), respectively, and pΩ1Ω2Ω3Ω1Ω2Ω3 with (13).IntegralI1 is equal to(27)I1=1λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i1=0∞1i1!Γi1+2μλi1+μ-1/2∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-11j2!Γj2+c∫0∞dΩ1Ω1-1-i1-μ+1/2+i1+2μ+j1+c-1e-λ/Ω1-Ω1/Ω01-ργi1+2μ,w2Ω1∫0Ω1dΩ2Ω2j1+j2+c-1e-Ω21+ρ/Ω01-ρ∫0Ω1dΩ3Ω3j2+c-1e-Ω3/Ω01-ρ.After using [12] for resolving the second and third integrals in (27), I1 becomes(28)I1=1λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i1=0∞1i1!Γi1+2μλi1+μ-1/2∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-11j2!Γj2+cΩ01-ρ1+ρj1+j2+cΩ01-ρj2+c∫0∞Ω1Ω1-1/2+μ+j1+c-1e-λ/Ω1-Ω1/Ω01-ργi1+2μ,w2Ω1γj1+j2+c,1+ρΩ1Ω01-ργj2+c,Ω1Ω01-ρ.After developing Gamma function defined by (17) and using [12] the following is obtained:(29)I1=1λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i1=0∞1i1!Γi1+2μλi1+μ-1/2∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-11j2!Γj2+cΩ01-ρ1+ρj1+j2+cΩ01-ρj2+c1i1+2μw2i1+4μ∑j3=0∞i1+2μ!i1+2μ+j3!w2j31j1+j2+c1+ρΩ01-ρj1+j2+c∑j4=0∞j1+j2+c!j1+j2+c+j4!1+ρΩ01-ρj41j2+c1Ω01-ρj2+c∑j5=0∞j2+c!j2+c+j5!1Ω01-ρj5λ+w2Ω01-ρ3+ρ-1/4-μ/2+j1+j2-j3/2+j4/2+j5/2+3c/2K-1/2-μ-i1+2j1+2j2-j3+j4+j5+3c2λ+w23+ρΩ01-ρ.IntegralI2 is equal to(30)I2=1λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i2=0∞1i2!Γi2+2μλi2+μ-1/2∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-11j2!Γj2+c∫0∞dΩ2Ω2-1-i2-μ+1/2+i2+2μ+j1+j2+c-1e-λ/Ω2-Ω21+ρ/Ω01-ργi2+2μ,w2Ω2∫0Ω2dΩ1Ω1j1+c-1e-Ω1/Ω01-ρ∫0Ω2dΩ3Ω3j2+c-1e-Ω3/Ω01-ρ.After using the procedure for solvingI1, I2 becomes(31)I2=1λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i2=0∞1i2!Γi2+2μλi2+μ-1/2∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-11j2!Γj2+cΩ01-ρj1+j2+2c1i2+2μw2i2+4μ∑j3=0∞i2+2μ!i2+2μ+j3!w2j31j1+c1Ω01-ρj1+c∑j4=0∞j1+c!j1+c+j4!1Ω01-ρj41j2+c1Ω01-ρj2+c∑j5=0∞j2+c!j2+c+j5!1Ω01-ρj5λ+w2Ω01-ρ3+ρ-1/4-μ/2-i2/2+j1+j2-j3/2+j4/2+j5/2+3c/2K-1/2-μ-i2+2j1+2j2-j3+j4+j5+3c2λ+w23+ρΩ01-ρ.After replacing the integralsI1 and I2 in (26), we obtain an expression for CDF of macrodiversity SC receiver output signal envelope:(32)Fww=21λ2μ-11Γc1-ρ2ρc-1Ω0c+2∑i1=0∞λi1+μ-1/2i1!Γi1+2μ∑j1=0∞ρΩ01-ρ2j1+c-11j1!Γj1+c∑j2=0∞ρΩ01-ρ2j2+c-11j2!Γj2+cw2i1+4μi1+2μ∑j3=0∞i1+2μ!i1+2μ+j3!w2j3Ω01-ρ1+ρj1+2j2+2c1j1+j2+c1+ρΩ01-ρj1+j2+c∑j4=0∞j1+j2+c!j1+j2+c+j4!1+ρΩ01-ρj41j2+c1Ω01-ρj2+c∑j5=0∞j2+c!j2+c+j5!1Ω01-ρj5λ+w2Ω01-ρ3+ρ-1/4-μ/2+j1+j2-j3/2+j4/2+j5/2+3c/2K-1/2-μ-i1+2j1+2j2-j3+j4+j5+3c2λ+w23+ρΩ01-ρ+Ω01-ρj1+j2+2c1j1+c1Ω01-ρj1+c∑j4=0∞j1+c!j1+c+j4!1Ω01-ρj41j2+c1Ω01-ρj2+c∑j5=0∞j2+c!j2+c+j5!1Ω01-ρj5λ+w2Ω01-ρ3+ρ-1/4-μ/2-i1/2+j1+j2-j3/2+j4/2+j5/2+3c/2K-1/2-μ-i1+2j1+2j2-j3+j4+j5+3c2λ+w23+ρΩ01-ρ.


## 5. Numerical Results

Probability density function of macrodiversity SC receiver output signal envelope versus SC receiver output signal envelope is plotted in Figure2 for several values ofk-μ fading severity parameter μ.Figure 2
Probability density function of microdiversity MRC receiver.In Figure3, cumulative distribution function of macrodiversity SC receiver output signal envelope versus SC receiver output signal envelope is plotted for differentk-μ fading severity parameter μ, Rician factor k, Gamma fading severity c, and correlation coefficient ρ.Figure 3
Cumulative distribution function of macrodiversity SC receiver.In Figure4, cumulative distribution function of macrodiversity SC receiver output signal envelope versus Rician factor k is plotted for differentk-μ fading severity parameter μ and correlation coefficient ρ.Figure 4
Cumulative distribution function of macrodiversity SC receiver output signal envelope versus Rician factork.


## 6. Conclusion

In this paper analysis of diversity system with three microdiversity MRC receivers and one macrodiversity SC receiver was done. At the inputs of microdiversity MRC receivers exist independentk-μ short term fading and correlated Gamma long term fading. Macrodiversity SC receiver reduces k-μ short term fading effects and microdiversity SC receivers reduce Gamma long term fading effects on system performances. For this system, the probability density function of macrodiversity SC receiver output signal envelope and the cumulative density function of macrodiversity SC receiver output signal envelope was calculated. Probability density function and cumulative density function are very important statistical characteristics. By using them we can calculate outage probability and average bit error probability.By settingk=0, thek-μ distribution reduces to Nakagami-m distribution and Rician distribution can be derived from k-μ distribution by setting for μ=1. Thek-μ distribution approximates Rayleigh distribution by setting k=0 and μ=1. System performances are on low level if we increase fading severity. With less k factor fading becomes of high severity.Increase of Riciank factor contributes to the decrease of cumulative density function. Cumulative density function of macrodiversity SC receiver output signal envelope decreases significantly faster for lower values of μ parameters, in comparison to higher value of parameter μ. With the increasing of correlation coefficient, the value of cumulative density function of macrodiversity SC receiver output signal envelope decreases. Signal envelope increases the value of cumulative density function growth and tends to 1. With increase of Rician k factor and the number of clusters μ, cumulative probability of signal grows faster.



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*Source: 9689586-2016-11-23.xml*