Fading models

Fading is caused by interference between two or more versions of transmitted signal which arrive at the receiver at slightly different times. These waves, called multipath waves, combine at the receiver antenna to give a resultant signal which can vary widely in amplitude and phase, depending on the distribution of the intensity and relative propagation time of the waves and the bandwidth of the transmitted signal.

In built-up urban areas, fading occurs because the height of the mobile antennas is well below the height of surrounding structures, so there is no single line-of-sight path to the base station.

The default values of Fading parameters in NetSim are as shown below Table-1.

Fading Model	Parameter	Value
Rayleigh	Scale Parameter	1
Nakagami	Shape parameter	1
Nakagami	Scale Parameter	1
Rician	Shape parameter	1
Rician	Scale Parameter	1

Table-1: Default values of Fading parameters

Nakagami Fading

In Nakagami fading, amplitude gain is Nakagami distributed. We know that when a variable is Nakagami distributed its square is Gamma distributed. Hence the power gain is Gamma distributed. The Gamma PDF, with shape parameter \(\alpha\) and scale parameter \(\beta\) is given by

\(f(x) = \frac{1}{\beta^{\alpha}\ \Gamma(\alpha)}x^{\alpha - 1}e^{- \left( \frac{x}{\beta} \right)}\) where \(\ \Gamma(\alpha) = \ \int_{0}^{\infty}{t^{\alpha - 1}e^{t}\ dt}\) is the Gamma function

NetSim uses a numerical method to generate a Gamma Random variate quickly based on [1]. The GUI takes shape parameter (m) and scale parameter (w) as input and a gamma distribution is generated with \(\alpha = m\) and \(\beta = \frac{w}{m}\). The values obtained in linear scale are then converted into dB. The default values for \(m\) and \(w\) are both 1.

Rayleigh Fading

In Rayleigh fading, the amplitude gain is Rayleigh distributed and the square of the amplitude is the power gain. Now we know that when a variable is Rayleigh distributed its square is exponentially distributed. The exponential distribution’s PDF is given as

\[f(x) = \lambda e^{- \lambda x}\]

An exponential random variate is generated in NetSim using

\[T = \ - \frac{\log_{e}{(R)}}{\lambda}\]

where R is a uniform random number on (0, 1). \(\lambda\) is \(\frac{1}{w}\) where \(w\) is the scale parameter that a user can set in the GUI. The default value of \(w\) is 1, and hence the default value of \(\lambda\) is also 1. The value of T is obtained in linear scale which is then converted to dB.

Rician Fading

In communications theory, Rician distributions are used to model scattered signals that reach a receiver by multiple paths. Depending on the density of the scatter, the signal will display different fading characteristics. Rician is circularly-symmetric bivariate normal random variable. The Rician distribution has a probability density function (pdf) is given by

\[f\left( x \middle| \nu,\sigma \right) = \frac{x}{\sigma^{2}}\exp\left( \frac{- \left( x^{2} + \nu^{2} \right)}{2\sigma^{2}} \right)I_{0}\left( \frac{x\nu}{\sigma^{2}} \right)\]

where, \(\nu\) is the mean, \(\sigma\) is the standard deviation and \(I_{0}(z)\) is the modified Bessel function of first kind with order zero. Rician fading also takes shape parameter (m) and scale parameter (w) as GUI inputs with default values set to 1. The shape parameter, defined as the ratio of the power contributions by line-of-sight path to the remaining multipaths is given by \(m = \frac{v^{2}}{2\sigma^{2}}\) and scale parameter, defined as the total power received in all paths as \(w = \nu^{2} + 2\sigma^{2}\). Using shape and scale parameters, we can determine the value of \(\nu\) and \(\sigma\) as \(\sigma = \sqrt{\frac{w}{m + 1}}\) and \(\nu = \sqrt{\frac{m + w}{2(m + 1)}}\). We thus, get fading loss (in linear scale) as \(R = X^{2} + Y^{2},\) from \(X\ \sim\ N(\nu cos\theta,\ \sigma^{2})\) and \(Y\ \sim\ N(\nu sin\theta,\ \sigma^{2})\) where \(\theta\ \)is any real number, \(N(\nu,\ \sigma^{2})\) stands for Normal Distribution with mean \(\nu\) and standard deviation \(\sigma\).