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Radio links are extensively used for voice and data communications at long distance. We analyze the radio propagation parameters that affect the received signal level on radio links in Rwanda and we determine the best path loss model for prediction of the received signal level. Various models of propagation and the mathematical expressions of path loss are described here in order to come to the prediction of those propagation effects. By analyzing data collected for two links of MTN Rwanda: Gahengeri-Kibungo and Gahengeri-Jali, we find that the best predicting model is the normal distribution.

The advanced accessibility and improved performance of radio links are needed for high quality signals transmission. According to [

In this work, we review the large scale fading models and their probability distributions; the other sections consist of a case study, results and discussions. The work ends with a conclusion.

The large-scale fading is characterized by average path loss and shadowing [

1) Free-space path loss model

This model takes into account the propagation along line-of-sight path between radio link terminals, transmitting and receiving antennas and is used for prediction of the received signal strength when transmitter and receiver have obvious, unobstructed LOS path between them [

where P_{t} is the transmitted signal power, P_{r} is the received signal power,

The path loss can be calculated by:

2) Log-distance path loss model

This model helps to forecast the path loss of a signal in a given environment. The general form of the path loss model is made by changing the free-space path loss with the path loss exponent n that varies with the environment.

The path loss exponent n is used to express as function of distance the average large-scale path loss for any transmitter-receiver separation. The path loss exponent n takes 2 as value for free space path loss model and it has a larger value in the presence of obstructions. This model is written as:

where d_{o} is a reference distance at which the path loss presents the characteristics of free space loss and d is the given path distance. The distance d_{o} = 1 km for a cellular system with a cell radius greater than 10 km and also d_{o} = 100 m or 1m for a macro cellular system with a cell radius of 1km or a microcellular with a very small radius [

3) Log-normal shadowing model

This model lets the receiver at the same distance d to have a different path loss, which varies with the random shadowing effects X_{σ}.

Environment | Path loss exponent, n |
---|---|

Free space | 2 |

Urban area cellular radio | 2.7 to 3.5 |

Shadowed urban cellular radio | 3 to 5 |

In building line-of-sight | 1.6 to 1.8 |

Obstructed in building | 4 to 6 |

Obstructed in factories | 2 to 3 |

The following equation is useful for modeling the path loss using log-normal shadowing:

From Equation (3), we have the extended equation of log-normal shadowing model:

where _{σ} is a zero-mean Gaussian distributed random variable (in dB) with standard deviation σ (also in dB) and _{0}. The same value can be determined through measurements.

The Q-function can be used to calculate the probability that the path loss PL(d) is less a particular value [

where the path loss, PL, can take l as its local value. The Q-function has the property which is written as:

The path loss is a random variable with a normal distribution in dB about the mean

This means that the probability that the path loss

As an example, from our data, let us see how to determine the path loss related to zero-mean normally distributed random variable

Using Q-function formula in (6), the probability P = 0.05 occurs for l = 1.645. The value l is given by:

In this equation, we use the approximate value of standard deviation,

Therefore, from Equation (9), the fade depths

1) Normal distribution

This distribution is applied to all real-valued continuous random variables. It is used as an approximation function to describe fluctuation of those variables around their single mean value. As the path loss is a random variable L, its probability density function is given by [

where

2) Lognormal distribution

This model describes statistically shadowing that can affects Line-of-sight (LOS). In this case, logarithmic decrease of received signal strength with respect to distance is estimated. The distribution of received signal strength is described by a normal distribution and is expressed as [

where

where

However, the log-normal distribution is the distribution of a positive variable whose logarithm has a normal distribution. It is possible therefore to represent directly the probability density function:

where

We conducted a survey of the proposed measured radio links from the network of MTN Rwanda in Eastern province of Rwanda. The collected data are for a period of six months for two existing radio links, Gahengeri-Kibungo and Gahengeri-Jali. For each radio link, the received power values were measured using Aviat Networks Portal. We collected the transmitted power values, the radio path lengths, the transmission frequencies, the antenna heights and the antenna gains. We consider the data for a period of six months, from January to June 2014.

The calculation of the difference between the transmitted power and the values of received powers will help to obtain the total path loss for each link.

1) Monthly average received power for each link

The received power is expressed in milliwatts. According to

From

For the link Gahengeli-Jali, the loss is very large in January.

Months | Gahengeri-Jali | Gahengeri-kibungo |
---|---|---|

Monthly average received power (in milliwatts) | ||

Jan | 2.69876E−06 | 3.05967E−05 |

Feb | 2.69536E−06 | 2.26206E−05 |

March | 2.60688E−06 | 3.08515E−05 |

April | 2.63211E−06 | 2.78625E−05 |

May | 2.84809E−06 | 2.75694E−05 |

June | 2.80083E−06 | 2.5337E−05 |

2) Total path loss observed during 6 months for each link

3) Comparison of normal and log-normal distributions for path loss

Calculating the difference in decibels (dB) for each measured received power, we obtain the total path loss for each radio link. We then determine the empirical pdf (measured pdf) of the path loss variation using numerical methods. To select the best model, we have to use the integral of square error (ISE) which is the sum of differences squared between measured pdf values and estimated pdf values.

The analysis of data was done by using MATLAB and the QI Macros 2014 in Excel software. This analysis will help to know the best path loss model.

1) Comparison of path loss models as function of path length, d.

2) Comparison of normal and log-normal distributions for path loss.

According to _{σ} (in dB) with standard deviation σ has to contributed to the path loss value. We have two graphs for two radio links because the standard deviation of this random variable depends on the distribution of the path loss for each link. The link that presents the higher log-normally distributed path loss value is Gahengeri-Jali (GJ) with long path length, as it is shown on

We observed that signal quality on radio links in Rwanda is mainly affected the attenuation of received signals due diffraction (because of highest points in Rwanda), reflection (because many mountains). High frequencies can sometimes contribute to this signal attenuation.

We presented models for determining the probability density function (pdf) of the path loss for the studied radio links as means of predicting the path loss variation. The models are based on the variation of the signal strength obtained from the difference between the transmitted power and the variation of the received power values during six months.

Combining the results of ISE and

This means that the large scale fading for radio propagation in Rwanda follow the normal distribution model. In fact, for two different links, one in the Eastern province of Rwanda (Gahengeri-Kibungo) and another in Central of Rwanda (Gahengeri-Jali), their best fitting model is the same (the normal distribution).

Radio links | ISE values | |
---|---|---|

Normal distribution | Log-normal distribution | |

Gahengeri-Jali | 2.24908E−09 | 9.09205E−09 |

Gahengeri-Kibungo | 1.01179E−09 | 3.49774E−08 |

About the validity of findings, the results obtained are reliable since the used data were collected from industry. These results may be applied in any field.

The radio propagation in Rwanda especially for large scale is characterized by path loss that may be due to reflection, diffraction, scattering and atmospheric conditions that cause the absorption of the signal and climatological conditions like rain that causes significant fluctuations of the signal and then leads to a low received power.

However, during the design of radio links in Rwanda, one should consider the path loss and the model by using the normal distribution to get a reliable radio link. In fact, the normal distribution was found to be the best fitting model to the measured probability density function.

Uwayezu Jean deDieu,Nsengiyumva Jean MarieVianney, (2016) Characterization and Modeling of Large-Scale Fading for Radio Propagation in Rwanda. Communications and Network,08,22-30. doi: 10.4236/cn.2016.81003