# Path Loss Measurements and Model Analysis in an Indoor Corridor Environment at 28 GHz and 38 GHz
## 3 Path Loss Measurements and Models
### Path Loss Propagation Models
#### 3.2.1 The Close-In (CI) Free Space Reference Distance Path Loss Prediction Model  
This is a model that has its primary principle on the anchor point and depends on the frequency in the free space. The free space path loss (FSPL) is present in the model parameter, which also depends on the frequency of the carrier propagating signal (f in GHz). The distance between the transmitter and the receiver (\(d\) in meters), as well as the chosen reference distance (\(d_{o}\)), are also important factors. The CI model has one parameter to be calculated in dB, i.e., PLE (\(n\)) [75; 42; 76]. The equation for the model is shown in (2).  
The reference distance of the CI model is 1 m, given by [75; 40]:  
\[P_{L}^{CI}\left(d\right)[dB]=FSPL(f,d_{o})[dB]+10.n.\log\left(\frac{d}{d_{o}} \right)+X_{\sigma SF}^{CI} \tag{2}\]  
For \(d\geq d_{o}\), where \(d_{o}=1m\)  
Where \(X_{\sigma SF}^{CI}\) is a zero mean gaussian random variable with a standard deviation \(\sigma\) in \(dB\).  
\[FSPLf,\ (d_{o})\ [dB]=10 log_{10}\left(\frac{4\pi d_{o}}{\lambda}\right) \tag{3}\]  
To determine the CI path loss model, the PLE \(n\) is found using the MMSE approach, which commiserates with the data measured by adopting a physical anchor location that represents the free space power transmitted from the antenna at the transmitter out to the reference distance of the CI \(d_{o}\). The reference distance \(d_{o}\)_of_\(1m\) is utilized in the mmWave CI model, which was proposed as a standard in [39] and [75]. Far-field radiation patterns (Fraunhofer distances) from high-gain directional antennas may be greater than 1 m from the antenna, but the CI path loss model may simply be converted to a 1 m reference distance by assuming that the far field begins at 1 m (even if it does not).  
\[FSPL(f,d_{o})[dB]=10 log_{10}\left(\frac{4\pi fd_{o}}{c}\right)^{2} \tag{4}\]  
\[FSPL(f,1m)[dB]=10 log_{10}\left(\frac{4\pi f}{c}\right)^{2} \tag{5}\]  
where \(c\) represents the speed of light.  
From (1), the shadow-fading random variable is  
\[X_{\sigma SF}^{CI}=PL^{CI}(d)[dB]-FSPL(f,1m)[dB]-10.n.\log(d) \tag{6}\]  
Let \(A=PL^{CI}(d)[dB]-FSPL(f,1m)[dB]\)_and_\(\mathrm{D}=10.n.\log(d)\)  
Therefore,  
\[X_{\sigma SF}^{CI}=-nD \tag{7}\]Then, the standard deviation of the random variable \(X_{\sigma SF}^{CI}\)_is_  
\[\sigma^{CI}=\sqrt{\sum\frac{{X_{\sigma}^{CI}}^{2}}{N}}{=}\sqrt{\sum\frac{{{{(A-nD)} }^{2}}}{N}} \tag{8}\]  
where \(N\) represents the number of measured path loss points.  
In order to obtain the PLE at the optimum value and \(\sigma^{CI}\), the partial differential of the numerator of Equation (8), with respect to the value of PLE, is equal to zero. Then,  
\[\frac{\partial{\left[{\sum{{(A-nD)}^{2}}}\right]}}{\partial{n}}=0 \tag{9}\]  
\[\sum 2D\ (nD-A)=0 \tag{10}\]  
\[2\sum D\ (nD-A)=2{\left({n\sum{D}^{2}}-\sum DA\right)} \tag{11}\]  
Therefore, from Equation (11), \(n=\frac{\sum DA}{\sum D^{2}}\)  
The minimum shadow fading (SF) standard deviation for the CI model is  
\[\sigma_{min}^{CI}=\sqrt{\frac{\sum{{{\left({A-D\frac{\sum{DA}}{\sum{D^{2}}}} \right)}^{2}}}}{N}} \tag{12}\]  
The closed-form solution value was processed through MATLAB, \(n\), and can be expressed using a Matrix formation;  
\[n=A^{T}{\left({D^{T}D}\right)}^{-1}D \tag{13}\]  
\[\sigma_{min}^{CI}=\sqrt{\frac{\sum{{{\left({A-A^{T}{{\left({D^{T}D}\right)}^{ -1}D}}\right)}^{2}}}}{N}} \tag{14}\]