## 3 Multichannel MAC Protocol with Directional Antenna for FANET
### Blind Rendezvous Algorithm with Directional Antenna  
The basic idea of the BR-DA algorithm is inspired by the operation of a circular clock with two hands moving at different walking speeds. The hands of the clock walk clockwise at their respective speeds and will surely meet each other at a specific time. Similarly, in a FANET, any pairwise main lobe of UAVs can be assimilated to these two hands in the clock, and the different time durations of the main lobe in each sector can be regarded as the different walking speeds of these two hands. Thus, the main lobe of UAVs can achieve rendezvous. We specify a time duration of \(M_{\text{T}}\)T and \(M_{R}\)T for the transmitter UAV and receiver UAV, respectively. \(M_{\text{T}}\) and \(M_{R}\) are different, coprime, and positive integers. T is the minimal slot duration to exchange discovery beacons between the pair of UAVs. The rendezvous period of the system composed of two UAVs is defined as the duration from the initial state to the last state. Note that during each period, the two UAVs search for all main lobe rendezvous cases according to the algorithms. Figure 7 illustrates the different state of the scenario of Section 2.3, and the arrows indicate the direction of the main lobe. It is important to note that the main lobe rendezvous state may not be achieved because full rendezvous cannot always be accomplished by different algorithms. **Theorem 1**. : _As two UAVs steer their antennas towards each other, the rendezvous period of the system is \(M_{T}M_{R}NT\)._  
Proof of Theorem 1.: The period of transmitter UAV and receiver UAV are \(M_{T}NT\) and \(M_{R}NT\) for scanning all directions, respectively. During each rendezvous period, the transmitter UAV and receiver UAV scan all directions and beamform towards the initial sectors they scan together. Hence, the rendezvous period of the system is the least common multiple of \(M_{T}NT\) and \(M_{R}NT\), i.e., \(M_{T}M_{R}NT\). \(\square\)  
**Theorem 2**. : \(M_{T}\) _and \(M_{R}\) need to satisfy the constraints \((M_{T}=1,M_{R}=N)\) and \((M_{T}=N,\)\(M_{R}=1)\) to achieve the theoretical supremum for clock synchronization discovery. \((M_{T}=1,\)\(M_{R}=N)\) presents \(M_{T}=1\) and \(M_{R}=N\) at the same time. The least delay of the full rendezvous algorithm is \(N^{2}T\) if and only if the above constraint is satisfied for clock synchronization discovery. Similarly, the least-worst-case (supremum) delay of blind rendezvous with clock synchronization is \(N^{2}T\)._  
Proof of Theorem 2.: The number of blind rendezvous cases for a pair of UAVs is \(N^{2}\). When \(N\) equals four, the number of blind rendezvous cases is 16, i.e., {(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4)}. Here, \((\cdot,\cdot)\) represents the sectors towards which the main lobe of a pair of UAVs beamform at the same slot. There is only one case in which they switch their antennas towards each other. If this pair of UAVs obtains full rendezvous diversity in a rendezvous period, the number of cases for this pair of UAVs is less than the rendezvous period, i.e., \(M_{T}M_{R}NT\geq N^{2}T\). Therefore, we can get \(M_{T}M_{R}\geq N\). In order to minimize the delay of the blind rendezvous algorithm, the rendezvous period of this pair of UAVs equals the number of cases, i.e., \(M_{T}M_{R}=N\). \(\square\)  
Figure 7: The rendezvous period of the system composed of two UAVs. (**a**) The initial state; (**b**) the main lobe rendezvous state; (**c**) the last state; and (**d**) the initial state of the next period. When the constraints \((M_{T}\
eq 1,M_{R}\
eq N)\) and \((M_{T}\
eq N,M_{R}\
eq 1)\) are applied to \(M_{T}M_{R}=N\), a full rendezvous diversity cannot be achieved. Moreover, under the constraints \((M_{T}\
eq 1,M_{R}\
eq N)\) and \((M_{T}\
eq N,M_{R}\
eq 1)\), we can obtain \(M_{T}\geq 2\) and \(M_{R}\geq 2\). In two continuous durations T, the sectors that the two UAVs beamform towards do not switch.