## 2 System Model and Problem Formulation
### MmWave Hybrid Beamforming  
In mmWave communications, large phase arrays are usually adopted to overcome the high propagation losses, and in combining with NOMA, mmWave beamforming is used to increase beam gain and serve multiple users. Usually, hybrid analog/digital beamforming is adopted in mmWave NOMA communications, where the precoding is performed in hybrid mode by combining the digital baseband precoding with an analog RF beamforming driven by a limited number of RF chains. This hybrid analog/digital beamforming is a cost-effective solution due to the use of massive antennas with limited RF chains. It can be easily implemented through the use of analog phase shifters together with the abilities of digital precoding, which allow the beams to be directed towards the desired user and remove inter-user interferences [23; 24].  
In particular, we consider the hybrid beamforming architecture at the BS with a number \(N_{RF}\) of RF chains exploiting NOMA in each RF chain and spatial division multiple access (SDMA) between RF chains [8], as shown in Figure 2.  
From the angle domain perspective, the knowledge of DoAs plays a fundamental role.  
In this paper, the UAVs' angle information are discovered by using the MUSIC algorithm implemented at the hybrid mmWave beamforming. We assume that the number of UAVs does not exceed the number of available RF chains, i.e., \(K\leq N_{RF}\).  
The received signal at the BS after RF and digital beamforming combining can be expressed as [19; 24]  
\[\mathbf{y}=\mathbf{D^{H}A^{H}Hs+D^{H}A^{H}w}, \tag{3}\]  
where \(\mathbf{H}_{N_{R}\times K}\) is the channel matrix (detailed in Equation (6)), \(N_{R}\) is the number of receiving antennas at the \(BS\), \(\mathbf{s}\) denotes the transmit signal, \(\mathbf{w}\) is the zero-mean independent and identically distributed (i.i.d.) Gaussian white noise vector with power \(\sigma_{w}^{2}\) and \(\mathbf{A}_{N_{R}\times N_{RF}}\), \(\mathbf{D}_{N_{RF}\times K}\) are the analog and digital beamforming matrices (see Equations (4) and (5)), respectively, with \(N_{RF}<N_{R}\).  
Figure 1: Multiple-UAV-relay-network-aided mmWave NOMA cellular system.  
In partially-connected hybrid mmWave beamforming, the antenna array with \(N_{R}\) elements can be organized into groups, called subarrays, and each subarray connected to one RF chain processes each received signal by a phase shifter. Then, all of them are added up as shown in Figure 2.  
Therefore, the analog matrix \(\mathbf{A}_{N_{R}\times N_{RF}}\) of Equation (3) is a diagonal matrix where  
\[v_{A,j}=\frac{1}{\sqrt{N_{R}}}[e^{i\phi_{j,1}},e^{i\phi_{j,2}}...e^{i\phi_{j,N _{R}}}] \tag{4}\]  
is the vector of subarray \(j\) with \(j=1,..N_{RF};\phi\) is chosen from a uniform distribution in the range of \([-\pi/2,\pi/2]\). The RF signal passes through \(N_{RF}\) parallel RF chains. It is down-converted and then the digital beamforming operation follows. The zero-forcing scheme can be used as a digital beamformer as  
\[\mathbf{D}=(\mathbf{H_{A}^{H}H_{A}})^{-1}\mathbf{H_{A}^{H}} \tag{5}\]  
where \(\mathbf{H_{A}}=\mathbf{A^{H}H}\) considers the actual channel matrix \(\mathbf{H}\) filtered by the analog beamforming matrix \(\mathbf{A}\).