# 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

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 in decibels (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 watts.
• ${\displaystyle P_{Rx_{dBm}}=10\log _{10}{\frac {P_{Rx}}{1mW}}}$ is the received power in dBm where ${\displaystyle {P_{Rx}}}$ is the received power in watts.
• ${\displaystyle PL_{0}}$ is the path loss in decibels (dB) at the reference distance ${\displaystyle d_{0}}$ calculated using the Friis free-space path loss model.
• ${\displaystyle {d}}$ is the length of the path.
• ${\displaystyle {d_{0}}}$ is the reference distance, usually 1 km (or 1 mile) for a large cell and 1 m to 10 m for a 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 decibels) caused by flat fading[citation needed]. In the case of no fading, this variable is 0. In the case of only shadow fading or slow fading, this random variable may have Gaussian distribution with ${\displaystyle \sigma \;}$ standard deviation in decibels, resulting in a log-normal distribution of the received power in watts. In the case of only fast fading caused by multipath propagation, the corresponding fluctuation of the signal envelope in volts may be modelled as a random variable with Rayleigh distribution or Ricean distribution[2] (and thus the corresponding gain in watts ${\displaystyle F_{g}\;=\;10^{\frac {-X_{g}}{10}}}$ 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. This can be convenient, because Power (Watts) is proportional to the square of amplitude. Squaring a Rayleigh-distributed random variable produces an Exponentially-distributed random variable. In many cases, exponential distributions are computationally convenient and allow direct closed-form calculations in many more situations than a Rayleigh (or even a Gaussian).

## 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