## 7 Results and Performance Evaluation
Therefore, this superposition of signals could be properly received with adequate equalizing techniques. It is important to note that these results shown in Figures 9 and 10 are independent of the frequency \(f_{c}\) used (900 MHz and 2.44 GHz) because, with an uniform \(AF\) (using same topology and number of motes), its gain is invariable with \(k\cdot l=2\pi f_{c}\cdot l=\frac{c\pi}{2}\), since \(l=\frac{\lambda}{4}=\frac{c}{4f_{c}}\). Figure 8: 16 TelosB motes deployed in a straight line to do beamforming in 2.44 GHz with \(l=\frac{\lambda}{4}\) mote spacing. However, our objective is to study the feasibility to obtain some CB gain in the transmission, and we find that the average/maximum likelihood of interfering signal at the desired angle (at 90\({}^{\circ}\)) is 2.817 (\(10\cdot log\) 2.817 = 4.49 dB gain) being N = 10 motes (\(\mu\) = 2.817, \(\sigma\) = 0.007) using \(f_{c}\) = 900 MHz and 2.816 with (\(\mu\) = 2.816, \(\sigma\) = 0.009) using \(f_{c}\) = 2.44 GHz, any case far from the ideal case. As stated previously for a uniform \(AF\), the gain is invariable with \(k\cdot l\), and, for that reason, both \(f_{c}\) get similar gains. In an ideal case, the CB gain coincides with the number of motes, thus the ideal CB gain would  
Figure 10: Histogram of CB signal deviation from the ideal uniform array pattern for a given pointing angle. Figure 9: Ideal array pattern for 10 motes in an uniform linear array with perfect zero delays pointing to 90\({}^{\circ}\) and one randomized trial (dashed line) of CB using the experimentally obtained PDF to generate the delays. be 10 (10 dB gain). However, some gain is achieved despite the synchronization errors. Moreover, we find that the same gain is achieved pointing to other angles. Figure 11 presents the average value of the total signal for angles between \(\theta=0^{\circ}\) and \(\theta=180^{\circ}\), for an \(M=10^{4}\) repetition scheme for 900 MHz and 2.44 GHz. By repeating the signal transmission \(M=10^{4}\) times, the average value of the \(AF\) at any direction \(\theta\) between \(0^{\circ}\) and \(180^{\circ}\) is also 2.817 (95% CI [2.816, 2.818]) \(f_{c}=900\) MHz and 2.816 (95% CI [2.815, 2.818]) \(f_{c}=2.44\) GHz, i.e., the repetition of the transmission eventually would reach any angular position relative to the more deployment with more power than a single more. A single mode would provide a signal strength of 1, whereas \(N=10\) motes with a sending repetition scheme would give 2.817 for both \(f_{c}\), but lower than a perfect uniform linear array. Similar results for both \(f_{c}\) were obtained with \(M=10^{3}\), (\(\mu_{CB_{gain}}=2.81\), \(\sigma_{CB_{gain}}=0.06\)) and \(M=5\times 10^{3}\) (\(\mu_{CB_{gain}}=2.80\), \(\sigma_{CB_{gain}}=0.03\)). Under these conditions, for example using \(f_{c}=2.44\) GHz and 10 motes in the uniform array (with 2.816 gain denoted as \(G_{tx}\)) transmitting 0 dBm (1 mW, denoted as \(P_{tx}\)), if the LEO satellite is 200 km above the earth, equipped with antennas of 30 dBi gain (denoted as \(G_{rx}\)) and receiver sensitivity typically of \(-118.5\) dBm [37], the received power at the satellite is \(-111.71\) dBm, given by \(\frac{P_{tx}}{4\pi R^{2}}\cdot G_{tx}\cdot\frac{\lambda^{2}}{4\pi}G_{rx}\)[17]. This received power is above the given sensitivity. Notice that we do not include the reflection effect (+ 3 dB) of the earth nor other attenuation effects in the transmission. In addition, we will consider the behavior of planar 2D array as shown in Figure 2b. In this case, \(AF_{2D}\) is given by (4), with \(N_{x}=10\) and \(N_{y}=5\), that is a \(10\times 5\) planar array with \(l_{x}=l_{y}=\frac{\lambda}{4}\). In this case, the gain in the ideal case is 50 (16.98 dB gain). Figure 12 shows the ideal case with perfect zero delays pointing to \(\phi=0^{\circ}\) and \(0^{\circ}<\theta<180^{\circ}\) compared against 10 CB trials, using the experimentally obtained PDF to generate the delays for 2.44 GHz.