## 2 System Model  
As shown in Figure 1, consider a downlink UAV communication system employing AN-GSDM, where the mobile UAV transmitter, [PERSON], is equipped with \(N_{t}\) antennas, and the legitimate user, [PERSON], is equipped with \(N_{r}\) (\(N_{r}<N_{l}\)) cooperative single-antenna receivers, which are located in different spatial directions and interconnected via optical fiber links. Additionally, [PERSON] activates \(N_{u}\) receivers for communication in a single time slot, where \(N_{u}<N_{r}\), meaning that only a subset of [PERSON]'s receivers participate in the transmission. Meanwhile, the eavesdropper, [PERSON], is equipped with \(N_{e}\) (\(N_{e}<N_{l}\)) cooperative single-antenna receivers, which are also interconnected via optical fiber links. Unlike the single-beam transmission in SDM, GSDM introduces receiver subset selection (RSS), which allows information to be transmitted through \(N_{u}\) activated beams selected from \(N_{r}\) available beams. This enables the transmission of \(k_{1}=\log_{2}C_{N_{r}}^{N_{u}}\) bits of information. Note that \(k_{1}\) might be a fractional value. In practical system design, it is typically rounded down to meet encoding requirements. However, for performance analysis, such as the security rate, the exact value of \(k_{1}\) can be used. The \(N_{u}\) activated antennas use traditional \(M\)-dimensional amplitude and phase modulation (APM), allowing for the transmission of \(k_{2}=N_{u}\log_{2}M\) bits of information. Furthermore, the total number of transmitted bits is \(k=k_{1}+k_{2}\). The multiple beams are used to transmit information in GSDM to Bob's multiple distributed receivers, which serve to enhance both the secrecy rate and transmission performance. The transmitted symbol vector \(\mathbf{s}_{l}^{m}\) is \(\mathbf{s}_{l}^{m}=\mathbf{I}_{l}\mathbf{b}_{m}\), where \(\mathbf{I}_{l}\) (\(l=1,2,\cdots,2^{k_{1}}\)), is a matrix consisting of \(N_{u}\) column vectors selected from the identity matrix \(\mathbf{I}_{N_{r}}\), corresponding to the  
Figure 1: The downlink UAV system employing AN-GSDM. \(N_{u}\) activated receivers. \(\mathbf{b}_{m}\)\(\left(m=1,2,\cdots,2^{k_{2}}\right)\) denotes a column vector constructed from \(N_{u}\) symbols selected from conventional APM symbols. Without loss of generality, assume that Alice is equipped with a uniform linear array (ULA), where the phase center is the geometric center of the array. The channel vector between Alice and the receiver located in the direction of angle \(\theta\) is then expressed as  
\[\mathbf{h}(\theta)=\frac{1}{\sqrt{N_{t}}}\left[e^{-j\left(\frac{N_{t}-1}{2} \right)\frac{2\pi}{\lambda}d\cos\theta},e^{-j\left(\frac{N_{t}-1}{2}-1\right) \frac{2\pi}{\lambda}d\cos\theta},\ldots,e^{j\left(\frac{N_{t}-1}{2}\right)\frac {2\pi}{\lambda}d\cos\theta}\right]^{H}, \tag{1}\]  
where \(\lambda\) is wavelength and \(d\) is is the antenna spacing of ULA. Let \(\Theta_{b}=\left(\theta_{b}^{1},\cdots,\theta_{b}^{N_{t}}\right)\) be the set of arrival angles for [PERSON]'s receivers, and let \(\Theta_{e}=\left(\theta_{e}^{1},\cdots,\theta_{e}^{N_{e}}\right)\) be the set of arrival angles for [PERSON]'s receivers. The legitimate downlink channel between [PERSON] and [PERSON], as well as the eavesdropping downlink channel between [PERSON] and [PERSON], can be represented, respectively, as follows:  
\[\mathbf{H}(\Theta_{b})=\left[\mathbf{h}\left(\theta_{b}^{1}\right),\cdots, \mathbf{h}\left(\theta_{b}^{N_{t}}\right)\right]^{H},\mathbf{H}(\Theta_{e}) =\left[\mathbf{h}\left(\theta_{e}^{1}\right),\cdots,\mathbf{h}\left(\theta_{ e}^{N_{e}}\right)\right]^{H}. \tag{2}\]  
To enhance the security, multi-beam precoding and AN are introduced to obscure the eavesdropper's ability.