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, calculated using the Friis free-space path loss model. Unit: Decibel (dB)

${\displaystyle {d}}$ is the length of the path.

${\displaystyle {d_{0}}}$ is the reference distance, usually 1 km (or 1 mile) for large cell and 1 m to 10 m for microcell [1].

${\displaystyle \gamma }$ 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. 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[2] (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.[3]

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

See also

• ITU Model for Indoor Attenuation
• Radio propagation model
• Young model

References

1. https://www.gaussianwaves.com/2013/09/log-distance-path-loss-or-log-normal-shadowing-model/
2. Julius Goldhirsh; Wolfhard J. Vogel. "11.4". Handbook of Propagation Effects for Vehicular and Personal Mobile Satellite Systems (PDF).
3. Wireless communications principles and practices, T. S. Rappaport, 2002, Prentice-Hall

Further reading

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