## 4 Results
### PCA with Antenna Radiation Patterns from Electromagnetic Simulations  
In the final set of simulations, a realistic four-element transmit antenna has been considered (Figure 7), as well as two omni antennas per receiver. In this case, the channel between an arbitrary transmit antenna (denoted as \(q\)) and an arbitrary receive antenna (denoted as \(u\)) for the \(l\)th multipath component is given according to 3 GPP specifications [17], as:  
\[h=\sqrt{\frac{P_{l}\sigma_{SF}}{M}}\sum_{m=1}^{M}\left( \begin{array}{c}\sqrt{ G_{BS}(\theta_{l,m,AoD})}\exp(j\left[k_{u}d_{q}\sin(\theta_{l,m,AoD})+\Phi_{l,m} \right])\times\\ \sqrt{G_{MS}(\theta_{l,m,AoA})}\exp(j(k_{u}d_{u}\sin(\theta_{l,m,AoA})))\end{array} \right) \tag{25}\]  
where \(j\) is the imaginary unit, \(P_{l}\) is the power of the \(l\)th path, \(\sigma_{SF}\) is the lognormal shadow fading, \(M\) is the number of sub-paths per path, \(\theta_{l,m,AoD}\) and \(\theta_{l,m,AoA}\) are the angles of departure (\(AoD\)) and arrival (\(AoA\)), respectively, for the \(m\)th subpath of the \(l\)th path, \(G_{BS}(\theta_{l,m,AoD})\) is the transmitter antenna gain for each array element and \(G_{MS}(\theta_{l,m,AoA})\) is the receiver antenna gain for each array element for the \(AoD\) and \(AoA\), respectively. Moreover, \(\Phi_{l,m}\) is the phase of the \(m\)th subpath of the \(l\)th path, uniformly distributed over \([0,2\pi]\). Finally, \(k_{u}\) is the wave number \(2\pi/\lambda\), where \(\lambda\) is the carrier wavelength in meters, \(d_{q}\) is the distance in meters of the transmit antenna element \(q\) from the reference (\(q=1\)) antenna and \(d_{u}\) and is the distance in meters of the receive antenna element \(u\) from the reference (\(u=1\)) antenna. The transmit antenna array consists of two elements, each one composed of two similar co-planar elements positioned in close proximity to each other and rotated one to the other, by an angle of \(90^{\circ}\), in order to obtain polarization diversity. The elements are fractal shaped, via a modified Hilbert curve configuration and were designed by the technique proposed in [20], the target being the dual frequency operation at ISM (2.4-2.489 GHz) and 5-6 GHz bands. The elements are semi-printed on the dielectric substrate RO3003 (\(\varepsilon_{r}=3\) and thickness \(h=0.812\) mm).The ground plane, having the shape shown in Figure 7, is on the back side of the substrate, extended o \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Power** & **Mean Number** & **Power** & **Mean Number** & **Power** & **Mean Number** \\
**Terms** & **of Repeats** (\(\overline{n}\)) & **Terms** & **of Repeats** (\(\overline{n}\)) & **Terms** & **of Repeats** (\(\overline{n}\)) \\ \hline
2 & 88.3652 & 14 & 95.4098 & 26 & 102.2866 \\
4 & 89.3452 & 16 & 96.6586 & 28 & 103.5060 \\
6 & 90.3492 & 18 & 97.8612 & 30 & 104.9076 \\
8 & 91.4246 & 20 & 98.9364 & 32 & 106.2758 \\
10 & 92.6398 & 22 & 100.0364 & 34 & 108.0206 \\
12 & 93.9894 & 24 & 101.0754 & 36 & 108.0000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mean number of repeats of the MSNR-PCA algorithm. Figure 7: **(a)** (left part) The four-element transmit array, size: 29.18 cm \(\times\) 6.15 cm, dielectric: RO3003, (\(\varepsilon_{r}=3\), \(h=0.812\) mm) (**b**) (right part) the incorporated EBG lattice: Size of patches: 16.8 mm \(\times\) 16.8 mm, distance between adjacent patches: 0.5 mm, size of EBG’s ground 36 mm \(\times\) 61.5 mm. antenna strip. This configuration, as proposed in [20], is not enough to ensure low values of all the correlation coefficients among the array's elements when two dual elements are arranged to compose a four-element array, because the four elements have (two by two) the same polarization.