Log-distance path loss model

File:Slow fading Log-distance.png

Log-distance path loss model is formally expressed as:

: $L=L\_\backslash text\{Tx\}-L\_\backslash text\{Rx\}=L\_0+10\backslash gamma\backslash log\_\{10\}\backslash frac\{d\}\{d\_0\}+X\_\backslash text\{g\}$

where

• $\{L\}$ is the total path loss in decibels (dB).
• $L\_\backslash text\{Tx\}=10\backslash log\_\{10\}\backslash frac\{P\_\backslash text\{Tx\}\}\{\backslash mathrm\{1~mW\}\}\; \backslash mathrm\{~dBm\}$ is the transmitted power level, and $P\_\backslash text\{Tx\}$ is the transmitted power.
• $L\_\backslash text\{Rx\}=10\backslash log\_\{10\}\backslash frac\{P\_\backslash text\{Rx\}\}\{\backslash mathrm\{1~mW\}\}\; \backslash mathrm\{~dBm\}$ is the received power level where $\{P\_\backslash text\{Rx\}\}$ 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.{{Cite web|url=https://www.gaussianwaves.com/2013/09/log-distance-path-loss-or-log-normal-shadowing-model/|title = Log Distance Path Loss or Log Normal Shadowing Model|date = 30 September 2013}}
• $\{d\}$ 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.
• $\backslash gamma$ is the path loss exponent.
• $X\_\backslash text\{g\}$ is a normal (Gaussian) random variable with zero mean, reflecting the attenuation (in decibels) caused by flat fading{{Citation needed|date=October 2011}}. 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 $\backslash sigma$ 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{{cite book|title=Handbook of Propagation Effects for Vehicular and Personal Mobile Satellite Systems|author=Julius Goldhirsh|author2=Wolfhard J. Vogel |url=http://vancouver.chapters.comsoc.org/files/2016/05/handbook.pdf|chapter=11.4}} (and thus the corresponding power gain $F\_\backslash text\{g\}=10^\{-X\_\backslash text\{g\}/10\}$ may be modelled as a random variable with exponential distribution).

This corresponds to the following non-logarithmic gain model:

: $\backslash frac\{P\_\backslash text\{Rx\}\}\{P\_\backslash text\{Tx\}\}=\backslash frac\{c\_0\; F\_\backslash text\{g\}\}\{d^\{\backslash gamma\}\}\; ,$

where

$c\_0=\{d\_0^\{\backslash gamma\}\}10^\{-L\_0/10\}$ 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 $F\_\backslash text\{g\}=10^\{-X\_\backslash text\{g\}/10\}$ is a stochastic process that reflects flat fading. In case of only slow fading (shadowing), it may have log-normal distribution with parameter $\backslash sigma$ 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 measurements of coefficients $\backslash gamma$ and $\backslash sigma$ in dB have shown the following values for a number of indoor wave propagation cases.Wireless communications principles and practices, T. S. Rappaport, 2002, Prentice-Hall

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

{{reflist}}

• {{cite book |last1=Seybold |first1=John S. |title=Introduction to RF Propagation |date=2005 |publisher=Wiley-Interscience |location=Hoboken, N.J. |isbn=9780471655961}}
• {{cite book |last1=Rappaport |first1=Theodore S. |title=Wireless Communications: Principles and Practice |date=2002 |publisher=Prentice Hall PTR |location=Upper Saddle River, N.J. |isbn=9780130995728 |edition=2nd}}

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Category:Radio frequency propagation model