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

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 [

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 [

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 [

The statistical behavior of signal in such systems can be described by different distributions as Rayleigh, Rice, Nakagami-

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) [

MRC diversity technique gives the best results. This technique effectively reduces the influence of

There are more works in open technical literature considering second order statistics of diversity systems. In [

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

Microdiversity MRC receiver reduces

The system that is being considered is shown in Figure

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.

Squared

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 [

Squared

Squared

The squared signal

Random variable

Relations between

Probability density function of microdiversity MRC receiver output signal envelope

One takes

After substituting (

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:

Signal envelopes powers at inputs in microdiversity receivers are correlated. Signal envelope powers

Probability density function of macrodiversity SC receiver output signal envelope

Functions

Integral

After using [

By using [

Integral

After the using procedure for solving

After replacing the integrals

Probability density function of microdiversity MRC receiver output signal envelope

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:

Cumulative density function of macrodiversity SC receiver output signal envelope

Functions

Integral

After using [

After developing Gamma function defined by (

Integral

After using the procedure for solving

After replacing the integrals

Probability density function of macrodiversity SC receiver output signal envelope versus SC receiver output signal envelope is plotted in Figure

Probability density function of microdiversity MRC receiver.

In Figure

Cumulative distribution function of macrodiversity SC receiver.

In Figure

Cumulative distribution function of macrodiversity SC receiver output signal envelope versus Rician factor

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 independent

By setting

Increase of Rician

The authors declare that they have no competing interests.