## 7 Results and Performance Evaluation  
To analyze the feasibility of CB using a set of N motes, we calculate the _Array Factor_ (AF) (or array pattern) [17] of the set using \(AF_{1D}\) and \(AF_{2D}\) given by Equatioins (3) and (4) seen in Section 3. We assume an isotropic (ominidirectional) antenna, such as the one embedded in TelosB motes transmitting 0 dBm [18]. In order to simulate random time delay as in a real scenario, as shown in Equation (5), we include these random delays in the offset--for example for \(AF_{1D}\) in Equation (3) as  
\[AF_{1D}=\sum_{i=0}^{N-1}e^{j_{i}(k\cdot cos(\theta)-\beta)+j_{i}\cdot\hat{s_{i} }}, \tag{6}\]  
where \(\hat{s_{i}}\) = phase offset due to the delay of mote \(i\). This phase offset can be expressed by \(\hat{s_{i}}=2\pi f_{c}d_{i}\) where \(d_{i}\) is the missynchronization time difference as shown in (5). Figure 7: Example of Meteor Burst Communications. First, we will analyze the behavior of a uniform linear array with \(AF_{\mathrm{LD}}\) in the ideal case given by Equation (3) and in the real scenario given by Equation (6). We will use \(f_{c}\) = 900 MHz and 2.44 GHz with \(N=10\) motes and \(l=\frac{\lambda}{4}\) mote spacing. The motes are expected to send simultaneously the same signal when they receive a beacon signal from a receiver located \(\theta=90^{\circ}\). The phase delay of the signal from each transmitter is thus associated to the signal traveling distance from each transmitter without any additional phase delay, in the ideal case, assuming zero progressive phase and perfect synchronism. The interference of the signals from each mote would give a maximum of the total signal in the direction orthogonal to the line where the motes are located, thus providing a CB gain to the transmission towards the receiver. The motes are at a fixed distance \(l\) along a straight line and the phase is set to zero at each transmitter. It is noted that this work is a feasibility study to implement these kinds of communications. Figure 8 shows 16 TelosB motes to do beamforming, operating at 2.44 GHz and separated \(l=\frac{\lambda}{4}\), approximately \(l=3.12\) cm. However, due to imperfect synchronism and location, the delay is not zero between motes and some randomized delay exists. In our model, we simulate the performance of the array by including this randomized delay obtained from experimental data, as shown in Section 5 in _Alternative 1_ using commercial motes. The histogram of the time delay (that includes the uncertainty both of time synchronization and location) was shown in Figure 5, where the average is \(\mu\) = 0 ns, \(\sigma\) = 165 ns (95% Confidence Interval (CI) [\(-\)7.3, +7.3] ns). These data are used to obtain the Probability Density Function (PDF) to generate random delays following a standard Monte Carlo Simulation (MC) [35] using Matlab [36]. The PDF fit is also shown in Figure 5.  
These time delays were introduced in the phase of each mote in the \(AF\) and the effect of misssynchronization is first seen in Figure 9. The plot of Figure 9 shows the ideal array pattern with perfect zero delays pointing to \(90^{\circ}\) and one randomized trial (dashed line) of CB using the above PDF to generate the delays. The ideal maximum (the ideal maximum is \(N=10\)) is not achieved in the desired angle (\(90^{\circ}\)). However, due to the random nature of these delays, we could ask to repeat the process of synchronization and emission M times, until, by chance, some of the CB transmissions reach the receiver. Following a standard MonteCarlo (MC) procedure, we simulate \(M\) trials where the motes continue sending the signal \(M\) times, \(M=10^{4}\). Figure 10 shows the histogram of the deviation of the CB signal from the ideal uniform \(AF\). The histogram shows a clear Rayleigh distribution, typical from a multipath scenario. Thus, it points to the argument that the superposition of signals from a CB deployment is equivalent, in terms of the receptor, to a single transmission in a multipath scenario, typical of a mobile communications environment.