Editing Free-space path loss (section)

== Free-space path loss formula ==
The free-space path loss (FSPL) formula derives from the [[Friis transmission equation|Friis transmission formula]].<ref name="Friis" />  This states that in a radio system consisting of a transmitting antenna transmitting radio waves to a receiving antenna, the ratio of radio wave power received <math>P_r</math> to the power transmitted <math>P_t</math> is:
:<math>\frac{P_r}{P_t} = D_t D_r \left( \frac{\lambda}{4 \pi d} \right)^2</math>
where
*<math>\ D_t</math> is the [[directivity]] of the transmitting antenna
*<math>\ D_r</math> is the [[directivity]] of the receiving antenna
*<math>\ \lambda</math> is the signal wavelength
*<math>\ d</math> is the distance between the antennas
The distance between the antennas <math>d</math> must be large enough that the antennas are in the [[Near and far field|far field]] of each other <math>\ d\gg\lambda</math>.<ref name="AEH">{{cite book|last1=Johnson|first1=Richard|title=Antenna Engineering Handbook|date=1984|publisher=McGraw-Hill, Inc.|location=New York, NY|isbn=0-07-032291-0|pages=1–12|edition=2nd}}</ref>
The free-space path loss is the loss factor in this equation that is due to distance and wavelength, or in other words, the ratio of power transmitted to power received assuming the antennas are [[isotropic radiator|isotropic]] and have no directivity (<math>D_t = D_r = 1</math>):<ref name="Whitaker">{{cite book
 | last1  = Whitaker
 | first1 = Jerry C. 
 | title  = The Electronics Handbook
 | publisher = CRC Press
 | date   = 1996
 | pages  = 1321
 | url    = https://books.google.com/books?id=DSHSqWQXm3oC&dq=f%22free+space+path+loss%22&pg=PA1321
 | isbn   =  9780849383458
 }}</ref>   
<math display="block">
\begin{align}
\mbox{FSPL} = \left ( \frac{4\pi d} \lambda \right )^2
\end{align}
</math>

Since the frequency of a radio wave <math>f</math> is equal to the [[speed of light]] <math>c</math> divided by the wavelength, the path loss can also be written in terms of frequency: 
<math display="block">
\begin{align}
\mbox{FSPL} = \left({4\pi df \over c}\right)^2
\end{align}
</math>

Beside the assumption that the antennas are lossless, this formula assumes that the [[polarization (waves)|polarization]] of the antennas is the same, that there are no [[multipath propagation|multipath]] effects, and that the radio wave path is sufficiently far away from obstructions that it acts as if it is in free space.  This last restriction requires an ellipsoidal area around the line of sight out to 0.6 of the [[Fresnel zone]] be clear of obstructions.  The Fresnel zone increases in diameter with the wavelength of the radio waves.  Often the concept of free space path loss is applied to radio systems that don't completely meet these requirements, but these imperfections can be accounted for by small constant power loss factors that can be included in the [[link budget]].