# Log-distance path loss model

The log-distance path loss model is a radio propagation model that predicts the path loss a signal encounters inside a building or densely populated areas over distance.

## Mathematical formulation

### The model

Log-distance path loss model is formally expressed as:

${\displaystyle PL\;=P_{Tx_{dBm}}-P_{Rx_{dBm}}\;=\;PL_{0}\;+\;10\gamma \;\log _{10}{\frac {d}{d_{0}}}\;+\;X_{g},}$

where

${\displaystyle {PL}}$ is the total path loss measured in Decibel (dB)
${\displaystyle P_{Tx_{dBm}}\;=10\log _{10}{\frac {P_{Tx}}{1mW}}}$ is the transmitted power in dBm, where
${\displaystyle P_{Tx}}$ is the transmitted power in watt.
${\displaystyle P_{Rx_{dBm}}\;=10\log _{10}{\frac {P_{Rx}}{1mW}}}$ is the received power in dBm, where
${\displaystyle {P_{Rx}}}$ is the received power in watt.
${\displaystyle PL_{0}}$ is the path loss at the reference distance d0. Unit: Decibel (dB)
${\displaystyle {d}}$ is the length of the path.
${\displaystyle {d_{0}}}$ is the reference distance, usually 1 km (or 1 mile).
${\displaystyle \gamma }$ is the path loss exponent.
${\displaystyle X_{g}}$ is a normal (or Gaussian) random variable with zero mean, reflecting the attenuation (in decibel) caused by flat fading[citation needed]. 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 ${\displaystyle \sigma \;}$ 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 ${\displaystyle F_{g}\;=\;10^{\frac {-X_{g}}{10}}}$ may be modelled as a random variable with Rayleigh distribution or Ricean distribution[1] (and thus the corresponding gain in Volts may be modelled as a random variable with Exponential distribution).

### Corresponding non-logarithmic model

This corresponds to the following non-logarithmic gain model:

${\displaystyle {\frac {P_{Rx}}{P_{Tx}}}\;=\;{\frac {c_{0}F_{g}}{d^{\gamma }}}}$

where

${\displaystyle c_{0}\;=\;{d_{0}^{\gamma }}10^{\frac {-PL_{0}}{10}}}$ is the average multiplicative gain at the reference distance ${\displaystyle d_{0}}$ from the transmitter. This gain depends on factors such as carrier frequency, antenna heights and antenna gain, for example due to directional antennas; and

${\displaystyle F_{g}\;=\;10^{\frac {-X_{g}}{10}}}$ is a stochastic process that reflects flat fading. In case of only slow fading (shadowing), it may have log-normal distribution with parameter ${\displaystyle \sigma \;}$ dB. In case of only fast fading due to multipath propagation, its amplitude may have Rayleigh distribution or Ricean distribution.

## Empirical coefficient values for indoor propagation

Empirical measurements of coefficients ${\displaystyle \gamma }$ and ${\displaystyle \sigma }$ in dB have shown the following values for a number of indoor wave propagation cases.[2]

Building Type Frequency of Transmission ${\displaystyle \gamma }$ ${\displaystyle \sigma }$ [dB]
Vacuum, infinite space 2.0 0
Retail store 914 MHz 2.2 8.7
Grocery store 914 MHz 1.8 5.2
Office with hard partition 1.5 GHz 3.0 7
Office with soft partition 900 MHz 2.4 9.6
Office with soft partition 1.9 GHz 2.6 14.1
Textile or chemical 1.3 GHz 2.0 3.0
Textile or chemical 4 GHz 2.1 7.0, 9.7
Office 60 GHz 2.2 3.92
Commercial 60 GHz 1.7 7.9

## References

1. ^ Julius Goldhirsh; Wolfhard J. Vogel. "11.4". Handbook of Propagation Effects for Vehicular and Personal Mobile Satellite Systems (PDF).
2. ^ Wireless communications principles and practices, T. S. Rappaport, 2002, Prentice-Hall