# LoS Theoretical and Experimental MIMO Study from 1-40 GHz in Indoor Environments
## 3 Path Loss and K-Factor  
The power delay profile (PDP), represented as \(P(\tau)\) is computed by averaging, spatially or temporally, the channel impulse response, \(h(t,\tau)\) over a local area:  
\[P(\tau)=\overline{\left|h(t,\tau)\right|^{2}}, \tag{1}\]  
where \(t\) is the time variable and \(\tau\) is the delay. Figure 3: Measured antenna pattern in dB for different frequencies. The channel impulse response \(h(t,\tau)\) is obtained by applying the inverse Fourier transform to the frequency response \(H(f)\). In this work, the PDP used to compute the delay parameters is obtained by applying the Hanning window in the frequency domain. This method improves the identification of the multipath components. The path loss is computed without applying the Hanning window so the energy is not altered. Figure 4 depicts the PDP for a given position (number 10 in Figure 2) using all measured bandwidth (see Table 2). In the horizontal axis, the distance in meters is shown by multiplying the delay \(\tau\) by the speed c. Due to the large bandwidth used, all main multipath components are clearly differentiated. From the power delay profile in time domain, or from the transfer function in frequency domain, the wideband PL can be computed either by integrating in delay domain or averaging in frequency, respectively. The measurements include the antenna effect; therefore, the antenna influence should be extracted from the measurements. The antenna gain depends on the frequency and the direction whereas the PL includes the effect in all frequencies of the bandwidth of all multipath components coming from different directions. Therefore, an estimation of the antenna effect has to be used; we computed this effect as the mean of the antenna gain in all frequencies and the mean in the azimuthal plane, the plane where the main multipath components and the direct ray are present. Thus, two values \(G_{\mathrm{tx}}\) and \(G_{\mathrm{rx}}\), are found for the transmitter and receiver. The path loss equation is:  
\[PL=-\big{\langle}\big{|}H(f)\big{|}_{2}\big{\rangle}_{f}+G_{rx}+G_{tx}=-\big{ \langle}\big{|}p(\tau)\big{|}_{2}\big{\rangle}_{\tau}+G_{rx}+G_{tx}, \tag{2}\]  
where \(\langle\ \rangle\) the expectation (average) operator, \(f\) is the frequency, and \(\tau\) is the delay. The PL of one specific component in the time domain can be extracted by applying the time gating technique. It is interesting to compare the total wideband PL with the LoS component PL as seen in Figure 5. For the sake of comparison, we have included in Figure 5 the PL evaluated using the Friis model which is observed to be in rather good agreement with the estimated LoS PL. Furthermore, due to the large bandwidth more energy is received from specular components, leading to a low K-factor as it will be shown later. Figure 4: The power delay profile (PDP) for position number 10 of Figure 2.  
In order to study different bands, we have focused on central frequencies 5 GHz, 15 GHz, 25 GHz and 35 GHz with a 10 GHz bandwidth. The following expression has been used for computing the 1-slope model (Floating-Intercept (FI) path loss). \[L=L_{0}+10n\times\log_{10}(d)+X_{o}\ (\mathrm{dB}), \tag{3}\]  
where:  
\(L_{0}\) (in dB) and \(n\) are adjustment parameters;  
\(d\) (in meters) is the distance between Tx and Rx, and  
\(X_{\sigma}\) (in dB), a Gaussian random variable with zero means and standard deviation \(\sigma\) (in dB). Figure 6 shows the relative received power mentioned bands and their corresponding 1-slope models. The slope parameters of the model can be seen in Table 3. The coefficient of determination R\({}^{2}\) indicates how linear the relation between variables is. We observe a nearly constant decay factor (\(n\)), lower than 2 in almost all cases. This low value corresponds to typical indoor values in small distances, similar to those measured in [22; 24; 25; 26; 27; 28; 33], as seen in Table 4. Indoor scenarios are rich multipath environments and reverberation effects usually make the decay factor to fall below 2.