Log-distance path loss model explained

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

Model

Log-distance path loss model is formally expressed as:

L = L _{Tx-L} _{Rx=L} _{0+10\gammalog} _{10} (d)/(d _{0}) + X _{g} where{L} is the total path loss in decibels (dB).

• $L\_\backslash text=10\backslash log\_\backslash frac\; \backslash mathrm$ is the transmitted power level, and

P _{Tx} is the transmitted power.

• $L\_\backslash text=10\backslash log\_\backslash frac\; \backslash mathrm$ is the received power level where

{P _{Rx} } is the received power.L _{0} is the path loss in decibels (dB) at the reference distance d _{0} . This is based on either close-in measurements or calculated based on a free space assumption with the Friis free-space path loss model.^[1] {d} is the length of the path.{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] \gamma is the path loss exponent.X _{g} is a normal (Gaussian) random variable with zero mean, reflecting the attenuation (in decibels) caused by flat fading. 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 \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 power gain $F\_\backslash text=10^$ may be modelled as a random variable with exponential distribution).

Corresponding non-logarithmic model

This corresponds to the following non-logarithmic gain model:

(P _{Rx})/(= P _{Tx}) (c _{0} F _{g})/(d ^{\gamma}) , where$c\_0=10^$ 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 $F\_\backslash text=10^$ 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. This can be convenient, because power 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

\gamma and \sigma in dB have shown the following values for a number of indoor wave propagation cases.^[3]

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

See also

• ITU model for indoor attenuation
• Radio propagation model
• Young model

Further reading

• Book: Seybold . John S. . Introduction to RF Propagation . 2005 . Wiley-Interscience . Hoboken, N.J. . 9780471655961.
• Book: Rappaport . Theodore S. . Wireless Communications: Principles and Practice . 2002 . Prentice Hall PTR . Upper Saddle River, N.J. . 9780130995728 . 2nd.

Notes and References

1. Web site: Log Distance Path Loss or Log Normal Shadowing Model. 30 September 2013.
2. Book: Handbook of Propagation Effects for Vehicular and Personal Mobile Satellite Systems. Julius Goldhirsh. Wolfhard J. Vogel . 11.4.
3. Wireless communications principles and practices, T. S. Rappaport, 2002, Prentice-Hall