# Shadowing
A [large-scale propagation models](large-scale%20propagation%20models.md) effect
## Physical Cause
Caused by random blocking by larger obstacles in the way. These can be trees, buildings etc.
Sometimes there is shadowing, other times there isn't.
## Effects
1. Variation in the short-term average of signal power (after [multipath fading](Multipath%20Fading.md) is averaged out)
2. The signal will vary even as you walk around in a circle of the same distance (basic path loss remains the same)
## How to obtain it from measurements
1. Average over $50-100\lambda$ (to remove [multipath](Multipath%20Fading.md) effects)
## Distances
10-100m
## Times
Doesn't really change with time - a static sort of loss.
## Maths
### Two types of effects
1. Reflection
2. Diffraction
These are abstract - we are going to approximate them so don't need to know the specifics!
Signal strength is given by:
$P_r=P_t\prod_{i=1}^{N_r}a_i\prod_{i=1}^{N_d}b_i$
in dB:
$P_r=P_t(dB)+\sum_{i=1}^{N_r}a_i(dB)+\sum_{i=1}^{N_d}b_i(dB)$
### Parameters
$a_i\rightarrow$ Signal attenuation due to the ith reflection
$b_i\rightarrow$ Signal attenuation due to the ith diffraction
$N_r$ and $N_d\rightarrow$ number of objects reflecting and diffracting the signal.
### Reformulation
$P_r=P_t(dB)+\underbrace{\sum_{i=1}^{N_r}a_i(dB)+\sum_{i=1}^{N_d}b_i(dB)}_{\sum_i\alpha(dB)}$
where $\sum_i\alpha(dB)$ represents a random and statistically independent attenuation.
It is a random variable since we don't know what objects are in the way.
The whole thing becomes Gaussian by the central limit theorem.
$A(dB) = \sum_i\alpha(dB) \xrightarrow[]{\text{C.L.T}} Gaussian$
So the received signal is
$\begin{align}P_r(dB) &= P_t(dB)+A(dB)\\
&= P_t(dB) +\underbrace{\mu_s(dB)}_{\text{Mean}}+X(dB)
\end{align}$
Where $A(dB)\sim N(\mu_s,\sigma^2)$ and $X(dB)\sim N(0,\sigma^2)$
Thus the path loss model (with shadowing effect) becomes
$PL(dB)=\underbrace{PL_{avg}(dB)}_{\mu_A}+X(dB)$
Where $PL_{avg}(dB)$ is the average path loss component and $\sigma$ typically ranges from 5-10dB.
Gaussian in the log domain.
### How to use
Need a mean and a variance (first and second order moments).
## Linear domain
Since X is normally distributed in dB scale, in linear scale we can define $A_S=10^{X/10}$ where $A_S$ is the attenuation factor due to shadowing.
We say this $A_S$ has a lognormal distribution.
So signal attenuation is also called lognormal shadowing.
## Coverage Percentage
$U(\gamma)=\frac{\int_0^{2\pi}\int_{d_0}^{R}Prob\left(P_r(r)\geq\gamma\right)rdrd\theta}{\pi(R^2-d_0^2)}$
## Verbose description
Local mean power fluctuates with a '[log-normal](log-normal.md)' distribution about the [area-mean](area-mean.md) power [^1]
Also uses [Friis Transmission Equation](Friis%20Transmission%20Equation.md).
Review [random variables](random%20variables.md)
## Uses
1. Power control design
2. 2nd order interference and [Tx](Transmitter.md) power analysis
1. You need more margin in your transmitter power!
2. You should give more power to account for the randomness of shadowing!
3. a more detailed link budget and cell coverage analysis
## References
1. [Wireless Communication](http://www.wirelesscommunication.nl/reference/chaptr03/shadow/shadow.htm#:~:text=Shadowing%20is%20the%20effect%20that,fluctuations%20due%20to%20multipath%20fading)
2. [Characterization of Slow and Fast Fading in V2I Channels for Smart Cities](../../../Spaces/University/Thesis/Support%20notes/Characterization%20of%20Slow%20and%20Fast%20Fading%20in%20V2I%20Channels%20for%20Smart%20Cities.md)
3. [TELE4652-lecture-03](../../../Spaces/University/TELE4652/Lectures/TELE4652-lecture-03.pdf)
## Footnotes
[^1]: this means that the log values of [local-mean](local-mean.md) power is distributed as a Gaussian normal.