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*1. Free Space Path Loss:-*

In telecommunication, free-space path loss (FSPL) is the loss in signal strength of an electromagnetic wave that would result from a line-of-sight path through free space (usually air), with no obstacles nearby to cause reflection or diffraction. It does not include factors such as the gain of the antennas used at the transmitter and receiver, nor any loss associated with hardware imperfections. A discussion of these losses may be found in the article on link budget. The FSPL is rarely used standalone, but rather as a part of the Friis transmission equation:

Free-space path loss is proportional to the square of the distance between the transmitter and receiver, and also proportional to the square of the frequency of the radio signal.

The equation for FSPL is

$FSPL=\left (\dfrac{4\pi d}{\lambda} \right )^2\\ \ \ \ \ \ \ \ \ \ \ \ \ =\left ( \dfrac{4\pi d f}{c} \right )^2$

where:

- $\lambda$ is the signal wavelength (in metres),
- $f$ is the signal frequency (in hertz),
* $d$ is the distance from the transmitter (in metres),
* $c$ is the speed of light in a vacuum, 2.99792458 × 108metres per second.
This equation is only accurate in the far field where spherical spreading can be assumed; it does not hold close to the transmitter.
***Free-space path loss in decibels***
A plot of the FSPL for a range of fixed frequencies. FSPL increases with increasing frequency.
$\begin{align
*} FSPL(dB)&=10log_{10}((\dfrac{4\pi}{c}df)^2)\ &=20log_{10}(\dfrac{4\pi}{c}df)\&=20log_{10}(d)+20log_{10}(f)+20log_{10}(\frac{4\pi}{c})\&=20log_{10}(d)+20log_{10}(f)-147.55 \end{align*}$ Where, the units are as before. For typical radio applications, it is common to find measured in units of MHz and in km, in which case the FSPL equation becomes $FSPL(dB)=20log_{10}(d)+20log_{10}(f)+ 32.45$ For $d,f$ in meters and kilohertz, respectively, the constant becomes . For $d,f$ in meters and megahertz, respectively, the constant becomes . For $d,f$ in kilometers and megahertz, respectively, the constant becomes . **Long Distance Path loss Model:** Log-distance path loss model is formally expressed as: $PL=P_{TxdBm}-P_{RxdBm}=PL_0+10\gamma \log_{10}\dfrac{d}{d_{0}}+X_g$ where $PL$ is the total path loss measured in Decibel (dB) $P_{TxdBm}=10log_{10}\dfrac{P_{Tx}}{1mW}$ is the transmitted power in dBm, where $P_{Tx}$is the transmitted power in watt. $P_{TxdBm}=10log_{10}\dfrac{P_{Rx}}{1mW}$ is the received power in dBm, where $P_{Rx}$ is the received power in watt. $PL_{0}$is the path loss at the reference distance *d*0. Unit: Decibel (dB) $d$ is the length of the path. $D_{0}$is the reference distance, usually 1 km (or 1 mile). $\gamma$ is the path loss exponent. $X_{g}$is a normal (or Gaussian) random variable with zero mean, reflecting the attenuation (in decibel) caused by flat fading. In case of no fading, this variable is 0. In case of only shadow fading or slow fading, this random variable may have Gaussian distribution with $\alpha$standard deviation in dB, resulting in log-normal distribution of the received power in Watt. In case of only fast fading caused by multipath propagation, the corresponding gain in Watts $F_{g}=10^{\frac{-X_{g}}{10}}$ may be modelled as a random variable with Rayleigh distribution or Ricean distribution ***Corresponding non-logarithmic model*** This corresponds to the following non-logarithmic gain model: $\dfrac{P_{Rx}}{P_{Tx}}=\dfrac{c_{o}F_{g}}{d^{\gamma}}$ where $c_{o}=d^{\gamma}_{o} 10^\frac{-l_{0}}{10}$ is the average multiplicative gain at the reference distance from the transmitter. This gain depends on factors such as carrier frequency, antenna heights and antenna gain, for example due to directional antennas; and $F_{g}=10^\frac{-X_{g}}{10}$ is a stochastic process that reflects flat fading. In case of only slow fading (shadowing), it may have log-normal distribution with parameter dB. In case of only fast fading due to multipath propagation, its amplitude may have Rayleigh distribution or Ricean distribution.