# 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[edit]

### The model[edit]

Log-distance path loss model is formally expressed as:

where

- is the transmitted power in dBm, where

- is the transmitted power in watt.

- is the received power in dBm, where

- is the received power in watt.

- is the length of the path.

- is the reference distance, usually 1 km (or 1 mile).

- is the path loss exponent.

- 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 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 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[edit]

This corresponds to the following non-logarithmic gain model:

where

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

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.

## Empirical coefficient values for indoor propagation[edit]

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

Building Type | Frequency of Transmission | [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[edit]

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

## Further reading[edit]

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