# 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.

## Applicable to / Under conditions

The model is used to predict the propagation loss for a wide range of environments

## Mathematical formulation

### The model

Log-distance path loss model is formally expressed as:

$PL\;=P_{Tx_{dBm}}-P_{Rx_{dBm}}\;=\;PL_0\;+\;10\gamma\;\log_{10} \frac{d}{d_0}\;+\;X_g,$

where

${PL}$ is the total path loss measured in Decibel (dB)
$P_{Tx_{dBm}}\;=10\log_{10} \frac{P_{Tx}}{1mW}$ is the transmitted power in dBm, where
$P_{Tx}$ is the transmitted power in watt.
$P_{Rx_{dBm}}\;=10\log_{10} \frac{P_{Rx}}{1mW}$ 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 d0. 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[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 $\sigma\;$ 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 $F_g\;=\;10^{\frac{-X_g}{10}}$ may be modelled as a random variable with Rayleigh distribution or Ricean distribution.[1]

### Corresponding non-logarithmic model

This corresponds to the following non-logarithmic gain model:

$\frac{P_{Rx}}{P_{Tx}}\;=\;\frac{c_0F_g}{d^{\gamma}}$

where

$c_0\;=\;{d_0^{\gamma}}10^{\frac{-L_0}{10}}$ is the average multiplicative gain at the reference distance $d_0$ from the transmitter. This gain depends on factors such as carrier frequency, antenna heights and antenna gain, for example due to directional antennas; and

$F_g\;=\;10^{\frac{-X_g}{10}}$ is a stochastic process that reflects flat fading. In case of only slow fading (shadowing), it may have log-normal distribution with parameter $\sigma\;$ 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 $\gamma$ and $\sigma$ in dB have shown the following values for a number of indoor wave propagation cases.[2]

Building Type Frequency of Transmission $\gamma$ $\sigma$ [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.
2. ^ Wireless communications principles and practices, T. S. Rappaport, 2002, Prentice-Hall

• Introduction to RF propagation, John S. Seybold, 2005, Wiley.
• Wireless communications principles and practices, T. S. Rappaport, 2002, Prentice-Hall.