## 3 Multichannel MAC Protocol with Directional Antenna for FANET
### Main Lobe Rendezvous Scheme-Based Blind Rendezvous Algorithm with Directional Antenna for FANET  
In Section 3.1, the BR-DA algorithm is used for the main lobe rendezvous of two UAVs. In this section, we introduce the BR-DA-FANET algorithm for the main lobe rendezvous of the FANET and the discovery beacon exchange scheme. In a FANET, each UAV cannot receive the mode of other UAVs. For neighbor discovery of a FANET, the main lobe of any two UAVs should be rendezvoused. Moreover, one UAV should be in transmission mode, and the other should be in reception mode simultaneously. A specific control sequence with a length of \(L\) bits is used to ensure the different modes for two neighboring UAVs in a FANET, as described in [15; 24]. We define a rendezvous-slot as an interval with duration \(\text{T}_{\text{rs}}\). \(\text{T}_{\text{rs}}\) equals \(N^{2}\text{T}\) and \(2N(N+1)\text{T}\) for clock synchronization discovery and clock asynchronization discovery, respectively. That is, in each rendezvous-slot, each UAV in a FANET can achieve blind rendezvous. If a rendezvous-slot is marked with a \"1\", the corresponding UAV operates in the transmission mode; otherwise, it operates in the reception mode. We can assign a specific sequence of 0's and 1's to each UAV to control the UAV's operation mode during the neighbor discovery process without centralized control. According to [15; 24], the specific control sequence consists of an \(l_{1}\)-bit segment of 1's, followed by the UAV's unique ID whose length is \(l_{2}\) bits and has an \(l_{0}\)-bit segment of 0's. The relationship of \(l_{0}\), \(l_{1}\), and \(l_{2}\) is expressed as:  
\[ \begin{cases}L=2l_{2}+1\\ l_{2}+1=l_{0}+l_{1}\end{cases}  \tag{6}\]  
In [15; 24], the authors have proved that any two different specific control sequences can guarantee that the two UAVs can operate in different modes for at least one-bit duration  
Figure 12: Antenna scan sequences for clock asynchronization discovery of UAV \(a\) and \(b\) according to Theorem 3.  
considering all possible cyclic rotations within a sequence of \(L\) consecutive bits. The BR-DA-FANET algorithm achieves main beam rendezvous for the FANET through the combination of specific control sequences and BR-DA. The least-worst-case discovery delay based on the BR-DA-FANET algorithm for clock synchronization discovery is expressed as (7). \[\mathrm{T_{Sy-BNetMAX}}{=}\mathrm{T_{rs}}\times L=LN^{2}\mathrm{T} \tag{7}\]  
The maximum number of UAVs in a FANET is \(U\). Therefore, the least-worst-case discovery delay of the BR-DA-FANET algorithm for clock synchronization discovery is expressed as (8), where \(\lceil\cdot\rceil\) represents the ceiling function. \[\mathrm{T_{Sy-BNetMAX}}{=}\mathrm{T_{rs}}\times L=LN^{2}\mathrm{T}{=}\left(2 \left\lceil\log_{2}^{U}\right\rceil+1\right)N^{2}\mathrm{T} \tag{8}\]  
**Theorem 4**. : \(\mathrm{2T_{rs}}\) _is the least duration for a rendezvous-slot to achieve main lobe blind rendezvous for a FANET. The least-worst-case discovery delay based on the BR-DA-FANET for clock asynchronization discovery of a FANET is \(2 LN(N+1)\mathrm{T}\)._  
Proof of Theorem 4.: There is a drift ranging from 0 to one \(\mathrm{T_{rs}}\) for the clock asynchronization discovery of a FANET. \(\square\)  
In Figure 13, Figure 13b represents Figure 13a with a drift ranging from 0 to 0.5 rendezvous-slot. When the drift ranges from 0 to 0.5 rendezvous-slot, the aligned time of any two UAVs' rendezvous-slot falls within the interval of 0.5\(\mathrm{T_{rs}}\) to \(\mathrm{T_{rs}}\). In order to achieve full rendezvous of the main lobe for a FANET, the aligned time of any two UAVs in the FANET must be greater than \(N(N+1)\mathrm{T}\). We define a rendezvous-slot as an interval with duration \(2N(N+1)\mathrm{T}\). Therefore, the main lobe rendezvous of a FANET can be achieved with drift which ranges from 0 to 0.5 rendezvous-slot. Figure 13: The picture for drift which ranges from 0 to 0.5\(\mathrm{T_{rs}}\).