## 2 Problem Formulation  
We consider a UAV-cellular communication scenario in which a UAV, as an aUE, transfers data over the uplink, while other UAVs simultaneously communicating with their serving BSs introduce interference with the target received uplink signal. The UAV's altitude is assumed to be above 40 m, in which case, according to 3 GPP Release 15, it has a LoS probability of 1 with the BS. To improve the received signal-to-interference-plus-noise ratio (SINR), the BS applies beamforming based on DoA estimation by first estimating the angle of arrival of the target signal and then executing beamforming. Here, we illustrate the sources of DoA estimation errors due to the UAV's mobility and the influence of these errors on the performance of beamforming algorithms. A brief discussion on such error has already been provided in Section 1. Here, we present a general introduction to DoA estimation errors. In a typical UAV-cellular communication scenario, multiple factors, including the angular resolution of the algorithms, the signal-to-noise ratio (SNR) and the angular spread effect, can affect the error range of DoA estimation algorithms. Moreover, additional DoA errors are caused by the UAV's mobility during the time interval between two DOA estimations. Suppose that two consecutive DoA estimations are performed at time instants \(t_{p_{n-1}}\) and \(t_{p_{n}}\), with an interval \(\Delta t=t_{p_{n}}-t_{p_{n-1}}\). During \(\Delta t\), the actual DoA of the incoming signal continuously varies, while the estimated DoA remains the same. Thus, at any time instant \(t_{m}\), \(t_{p_{n-1}}\leq t_{m}\leq t_{p_{n}}\), a DoA estimation error exists, which is equal to the difference between the estimated angle and the actual angle:  
\[e(t_{m})=|\gamma_{e}-\eta(t_{m})|, \tag{1}\]  
where \(e(t_{m})\) denotes the estimation error at \(t_{m}\) and \(\eta(t_{m})\) and \(\gamma_{e}\) are the corresponding actual and estimated DoA values, respectively. In addition to a longer DoA estimation interval, a higher UAV speed or a reduced cell radius in the 5G system will also cause this DoA estimation error to increase [20; 25]. The gain achieved with classical nonrobust beamforming decreases dramatically with even a minor DoA error [26; 27]. Although robust beamforming methods have been developed to suppress such sensitivity, the optimal gains of all these algorithms degrade to various extents as the DoA error increases. We take the widely studied diagonal loading (DL) algorithm for robust beamforming as an example to illustrate this effect. In this algorithm, the beamforming weight vector, \(w\), is obtained as follows:  
\[\mathbf{w}=\frac{(\mathbf{R}+\sigma\mathbf{I}_{L})^{-1}\mathbf{s}(\theta)}{ \mathbf{s}^{\mathrm{H}}(\theta)(\mathbf{R}+\sigma\mathbf{I}_{L})^{-1}\mathbf{ s}(\theta)}, \tag{2}\]  
where \(\mathbf{I}_{L}\) is an identity matrix; \(\mathbf{s}(\theta)\) denotes the steering vector of the target signal and \(\theta\) is the estimated DoA value; \(\mathbf{R}\) is an \(L\times L\) array correlation matrix, the elements of which represent the correlations between the received signals of the array antennas. The coefficient \(\sigma\), known as the DL factor, modifies \(\mathbf{R}\) by enhancing the effect of noise, forcing the beamformer to mitigate noise and thus weakening its ability to suppress interference. In this way, the beamformer is made less sensitive to DoA errors. Over a span of many years and up through recent times, a large number of algorithms have been proposed for mapping the DL factor with a certain DoA error range to achieve optimal performance [26; 28; 29; 30]. However, simply finding the most suitable DL factor will not result in optimal gains since the DoA error range places an upper limit on the achievable gain. A plot of the SINR output of a DL beamformer versus the DL factor is shown in Figure 3, where the input SNR is 10 dB and the number of antenna elements is 8.