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Log-distance path loss model
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<h2 id="mathematical-formulation">Mathematical formulation</h2> <h3>Model</h3> <p>Log-distance path loss model is formally expressed as: </p> L = L Tx − L Rx = L 0 + 10 γ log 10 d d 0 + X g {\displaystyle L=L_{\text{Tx}}-L_{\text{Rx}}=L_{0}+10\gamma \log _{10}{\frac {d}{d_{0}}}+X_{\text{g}}} <p>where </p> <ul><li> L {\displaystyle {L}} is the total <a href="/facts/Path_loss/I34lqCYk">path loss</a> in <a href="/facts/Decibel/0LyxAPXh">decibels</a> (dB).</li> <li> L Tx = 10 log 10 P Tx 1 m W d B m {\textstyle L_{\text{Tx}}=10\log _{10}{\frac {P_{\text{Tx}}}{\mathrm {1~mW} }}\mathrm {~dBm} } is the transmitted power <a href="/facts/Level_(logarithmic_quantity)/0Yt12Pfr">level</a>, and P Tx {\displaystyle P_{\text{Tx}}} is the transmitted power.</li> <li> L Rx = 10 log 10 P Rx 1 m W d B m {\textstyle L_{\text{Rx}}=10\log _{10}{\frac {P_{\text{Rx}}}{\mathrm {1~mW} }}\mathrm {~dBm} } is the received power level where P Rx {\displaystyle {P_{\text{Rx}}}} is the received power.</li> <li> L 0 {\displaystyle L_{0}} is the <a href="/facts/Path_loss/I34lqCYk">path loss</a> in decibels (dB) at the reference distance d 0 {\displaystyle d_{0}} . This is based on either close-in measurements or calculated based on a free space assumption with the Friis <a href="/facts/Free-space_path_loss/wQccyFS3">free-space path loss</a> model.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li> <li> d {\displaystyle {d}} is the length of the path.</li> <li> d 0 {\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.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li> <li> γ {\displaystyle \gamma } is the <a href="/facts/Path_loss/I34lqCYk">path loss</a> exponent.</li> <li> X g {\displaystyle X_{\text{g}}} is a <a href="/facts/Normal_random_variable/UapjjPyQ">normal (Gaussian) random variable</a> with zero <a href="/facts/Mean/swcEd4Pg">mean</a>, reflecting the attenuation (in decibels) caused by <a href="/facts/Flat_fading/RsDwBbdz">flat fading</a>. In the case of no fading, this variable is 0. In the case of only <a href="/facts/Shadow_fading/RsDwBbdz">shadow fading</a> or <a href="/facts/Slow_fading/RsDwBbdz">slow fading</a>, this random variable may have <a href="/facts/Gaussian_distribution/UapjjPyQ">Gaussian distribution</a> with σ {\displaystyle \sigma } <a href="/facts/Standard_deviation/qSug9BRl">standard deviation</a> in decibels, resulting in a <a href="/facts/Log-normal_distribution/V2RXmmlE">log-normal distribution</a> 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 <a href="/facts/Rayleigh_distribution/vLFh4zHr">Rayleigh distribution</a> or <a href="/facts/Ricean_distribution/E9tY1XNJ">Ricean distribution</a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> (and thus the corresponding power gain F g = 10 − X g / 10 {\textstyle F_{\text{g}}=10^{-X_{\text{g}}/10}} may be modelled as a random variable with <a href="/facts/Exponential_distribution/yY3EdV0u">exponential distribution</a>).</li></ul> <h3>Corresponding non-logarithmic model</h3> <p>This corresponds to the following non-logarithmic gain model: </p> P Rx P Tx = c 0 F g d γ , {\displaystyle {\frac {P_{\text{Rx}}}{P_{\text{Tx}}}}={\frac {c_{0}F_{\text{g}}}{d^{\gamma }}},} <p>where c 0 = d 0 γ 10 − L 0 / 10 {\textstyle c_{0}={d_{0}^{\gamma }}10^{-L_{0}/10}} is the average multiplicative gain at the reference distance d 0 {\displaystyle d_{0}} from the transmitter. This gain depends on factors such as <a href="/facts/Carrier_frequency/ae98pBlY">carrier frequency</a>, antenna heights and antenna gain, for example due to directional antennas; and F g = 10 − X g / 10 {\textstyle F_{\text{g}}=10^{-X_{\text{g}}/10}} is a <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic process</a> that reflects <a href="/facts/Flat_fading/RsDwBbdz">flat fading</a>. In case of only slow fading (shadowing), it may have <a href="/facts/Log-normal/V2RXmmlE">log-normal</a> distribution with parameter σ {\displaystyle \sigma } dB. In case of only <a href="/facts/Fast_fading/RsDwBbdz">fast fading</a> due to <a href="/facts/Multipath_propagation/dJFXvAbW">multipath propagation</a>, its amplitude may have <a href="/facts/Rayleigh_distribution/vLFh4zHr">Rayleigh distribution</a> or <a href="/facts/Ricean_distribution/E9tY1XNJ">Ricean distribution</a>. This can be convenient, because power is proportional to the square of amplitude. Squaring a Rayleigh-distributed random variable produces an <a href="/facts/Exponential_distribution/yY3EdV0u">exponentially distributed</a> 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). </p> <h2 id="empirical-coefficient-values-for-indoor-propagation">Empirical coefficient values for indoor propagation</h2> <p>Empirical measurements of coefficients γ {\displaystyle \gamma } and σ {\displaystyle \sigma } in dB have shown the following values for a number of indoor wave propagation cases.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <table><tbody><tr><th>Building type</th><th>Frequency of transmission</th><th> γ {\displaystyle \gamma } </th><th> σ {\displaystyle \sigma } [dB]</th></tr><tr><td>Vacuum, infinite space</td><td></td><td>2.0</td><td>0</td></tr><tr><td>Retail store</td><td>914 MHz</td><td>2.2</td><td>8.7</td></tr><tr><td>Grocery store</td><td>914 MHz</td><td>1.8</td><td>5.2</td></tr><tr><td>Office with hard partition</td><td>1.5 GHz</td><td>3.0</td><td>7</td></tr><tr><td>Office with soft partition</td><td>900 MHz</td><td>2.4</td><td>9.6</td></tr><tr><td>Office with soft partition</td><td>1.9 GHz</td><td>2.6</td><td>14.1</td></tr><tr><td>Textile or chemical</td><td>1.3 GHz</td><td>2.0</td><td>3.0</td></tr><tr><td>Textile or chemical</td><td>4 GHz</td><td>2.1</td><td>7.0, 9.7</td></tr><tr><td>Office</td><td>60 GHz</td><td>2.2</td><td>3.92</td></tr><tr><td>Commercial</td><td>60 GHz</td><td>1.7</td><td>7.9</td></tr></tbody></table> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/ITU_model_for_indoor_attenuation/ABt9myIT">ITU model for indoor attenuation</a></li> <li><a href="/facts/Radio_propagation_model/80y1JV4M">Radio propagation model</a></li> <li><a href="/facts/Young_model/pz6oDo1y">Young model</a></li> <li><a href="/facts/Path_loss/I34lqCYk">Path loss</a></li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Seybold, John S. (2005). <i>Introduction to RF Propagation</i>. Hoboken, N.J.: Wiley-Interscience. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780471655961.</li> <li>Rappaport, Theodore S. (2002). <i>Wireless Communications: Principles and Practice</i> (2nd ed.). Upper Saddle River, N.J.: Prentice Hall PTR. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780130995728.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Log Distance Path Loss or Log Normal Shadowing Model". 30 September 2013. <a href="https://www.gaussianwaves.com/2013/09/log-distance-path-loss-or-log-normal-shadowing-model/" target="_blank">https://www.gaussianwaves.com/2013/09/log-distance-path-loss-or-log-normal-shadowing-model/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>"Log Distance Path Loss or Log Normal Shadowing Model". 30 September 2013. <a href="https://www.gaussianwaves.com/2013/09/log-distance-path-loss-or-log-normal-shadowing-model/" target="_blank">https://www.gaussianwaves.com/2013/09/log-distance-path-loss-or-log-normal-shadowing-model/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Julius Goldhirsh; Wolfhard J. Vogel. "11.4". Handbook of Propagation Effects for Vehicular and Personal Mobile Satellite Systems (PDF). <a href="http://vancouver.chapters.comsoc.org/files/2016/05/handbook.pdf" target="_blank">http://vancouver.chapters.comsoc.org/files/2016/05/handbook.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Wireless communications principles and practices, T. S. Rappaport, 2002, Prentice-Hall <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>