# Path loss
Distance dependant attenuation [^wireless-lect]
## Cause
Reduction in power density of an EM wave as it propagates through space.
## Common distances
100m to 1km
[^slow-and-fast-fading]
## General Path loss definition
The linear path loss of a channel is the ratio of the transmit to received power:
$P_L=\frac{P_T}{P_R}$
It's easier to consider this in the log domain.
$PL(dB) = 10\log_{10}\left(\frac{P_T}{P_R}\right)(dB)$
Which is non negative (if received power is less than transmitted power).
see [Friis Transmission Equation](Friis%20Transmission%20Equation.md).
The received signal power in dB:
$P_L = 20\log_{10}(d)+20\log_{10}(f)-K$
where $K=10\log_{10}\left(\frac{G_tG_rc^2}{16\pi^2}\right)$
### Assumptions
1. The transmitter and receiver have a clear [line of sight](Line%20of%20sight.md)
1. No other sources of impairment!
## Where is free space path loss seen?
Satellite systems and microwave systems undergo free space propagation.
## Description
The distance dependent attenuation
Happens due to the inverse square law - power is being shared across larger areas.
For free space, 
But the path loss exponent changes it (with lots of buildings etc.)
Mostly values of n from 3-4
## Special Case
Indoors - n <2
Long walls contain a signal - pipelining.
## Measuring path loss exponent
You take many measurements and fit a curve to the log of the distance:
The inverse square law only holds in free space.
## Uses
1. System planning
2. Cell coverage planning
3. [Link Budget](Link%20Budget.md)
## Example
Determine the isotropic free space loss at 4GHz for the shortest path to a geosychronous satellite from earth (35,863km) for unit gain antennas.
$P_L=20\log_{10}(4\times10^9)+20\log_{10}(35.863\times10^6)-147.56=195.6[dB]$
If the earth station transmits at 250W, the received power is -141dBm[^2]
## More realistic equation (reference distance)
$d_0$ is the received power reference point.
You find the path loss at the reference point first ($P_L(d_0)$) then find free space path loss at any position using:
$P_L(d)=P_L(d_0)\left(\frac{d}{d_0}\right)^n$
where n is 2 for free space, and other n take into account multipath and shadowing effects.
or in dB:
$P_L(d)\ [dB]= P_L(d_0)\ [dB]+10n\log_{10}(\frac{d}{d_0})$
### Definition of path loss at the received power reference point
## Path loss exponent
### Typical Values
| Environment |Path Loss Exponent |
|:--:|:--:|
| In building [line of sight](Line%20of%20sight.md) [^3]|1.6-1.8 |
|Free space|2|
|Urban area cellular|2.7-3.5|
|Shadowed urban area|3-5|
## For n=2
A 6dB drop of received signal strength per doubling of distance
## More accurate models
1. Models capturing effects of antenna heights
2. Indoor propagation models capturing wall and floor attenuation
3. Outdoor models capturing rain and weather
4. Carrier frequency dependent models
[Okumura-Hata Model](Okumura-Hata%20Model.md)
[Middle-Zone Model](Middle-Zone%20Model.md)
# References
1. [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)
2. [TELE4652 Lecture 3](../../Spaces/University/TELE4652/Lectures/TELE4652-lecture-03-mobile-radio-propagation-models.pdf)
3. [UNSW Lecture 1 on Wireless Fading Channel Modeling and Estimation](../../../Spaces/University/Thesis/docs/Videos/UNSW%20Lecture%201%20on%20Wireless%20Fading%20Channel%20Modeling%20and%20Estimation.mp4)
# Footnotes
[^2]: $10\log_{10}(250/0.001)-195.6 = -141 [dBm]$
[^3]: The ceiling, walls and floor act as a waveguide to help the signal propagate