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Log normal Shadowing
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The Model (Log-distance Path Loss ) in equation 2 does not consider the fact that the surrounding environmental clutter may be vastly different at two different locations having the same T-R separation.

This leads to measured signals which are vastly different than the average value predicted by Equation 2 in Log-distance Path Loss model.

Measurements have shown that at any value of d , the path loss $P L(d)$ at a particular location is random and distributed log-normally about the mean distance-dependent value. That is

$P L(d)[d B]=\overline{P L}(d)+X_{\sigma}=\overline{P L}\left(d_{0}\right)+10 n \log \left(\frac{d}{d_{0}}\right)+X_{\sigma}-----(1)$

and

$P_{r}(d)[d B m]=P_{t}[d B m]-P L(d)[d B] \quad \text { (antenna gains included in P L(d) )}-----(2)$

where $X_{\sigma}$ is a zero-mean Gaussian distributed random variable with standard deviation $\sigma$.

The log-normal distribution describes the random shadowing effects which occur over a large number of measurement locations which have the same T-R separation, but have different levels of clutter on the propagation path.

This phenomenon is referred to as log -normal shadowing. Simply put, log-normal shadowing implies that measured signal levels at a specific T-R separation have a Gaussian (normal) distribution about the distance-dependent mean of Equation 2 in Log-distance Path Loss model, where the measured signal levels have values in $\mathrm{dB}$ units.

The standard deviation of the Gaussian distribution that describes the shadowing also has units in $\mathrm{dB}$ . Thus, the random effects of shadowing are accounted for using the Gaussian distribution which lends itself readily to evaluation.

In practice, the values of $n,$ and $\sigma$ . are computed from measured data, using linear regression such that the difference between the measured and estimated path losses is minimized in a mean square error sense over a wide range of measurement locations and T-R separations.

The value of $\overline{P L}\left(d_{0}\right)$ in eqn (1) is based on either close-in measurements or on a free space assumption from the transmitter to $d_{0}$ .

An example of how the path loss exponent is determined from measured data follows. Figure below illustrates actual measured data in several cellular radio systems and demonstrates the random variations about the mean path loss (in $\mathrm{dB}$ ) due to shadowing at specific T-R separations.

since $P L(d)$ is a random variable with a normal distribution in $\mathrm{dB}$ about the distance- dependent mean, so is $P_{r}(d),$ and the $Q,$ -function or error function (erf) may be used to determine the probability that the received signal level will exceed (or fall below) a particular level. The $Q,$ -function is defined as

$Q(z)=\frac{1}{\sqrt{2 \pi}} \int_{z}^{\infty} \exp \left(-\frac{x^{2}}{2}\right) d x=\frac{1}{2}\left[1-\operatorname{erf}\left(\frac{z}{\sqrt{2}}\right)\right]-----(3)$

where,

$Q(z)=1-Q(-z)-----(4)$

The probability that the received signal level (in $\mathrm{dB}$ power units) will exceed a certain value $\gamma$ can be calculated from the cumulative density function as

$\operatorname{Pr}\left[P_{r}(d)\gt\gamma\right]=Q\left(\frac{\gamma-\overline{P_{r}(d)}}{\sigma}\right)-----(5)$

Similarly, the probability that the revived signal level will be below $\gamma$ is given by

$\operatorname{Pr}\left[P_{r}(d)\lt\gamma\right]=Q\left(\frac{\overline{P_{r}(d)}-\eta}{\sigma}\right)-----(6)$

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