## 2 System Model
### The Proposed Secure Beam Hybrid Precoding Algorithm: GSVD-Sparsity  
In the fully digital architecture, the optimal secure beam can be formed by solving the precoder/combiner pair \((\mathbf{F}_{opt},\mathbf{W}_{opt})\) based on GSVD. However, in the mmWave hybrid precoding architecture, matrix operations are required to decompose \(\mathbf{F}_{opt}\) (\(\mathbf{W}_{opt}\)) into \(\mathbf{F}_{RF}\) and \(\mathbf{F}_{BB}\) (\(\mathbf{W}_{RF}\) and \(\mathbf{W}_{BB}\)). However, in general mmWave communication systems, the number of RF chains is not equal to the number of transmit (receive) antennas. Typically, the number of RF chains is much smaller than the number of transmit (receive) antennas. In this case, the decomposition problem becomes an optimization problem, which requires \(\mathbf{F}_{RF}\mathbf{F}_{BB}\) (\(\mathbf{W}_{RF}\mathbf{W}_{BB}\)) to infinitely approach \(\mathbf{F}_{opt}\) (\(\mathbf{W}_{opt}\)), i.e.,  
\[ \begin{split}\mathbf{F}_{RF}\mathbf{F}_{BB}& \rightarrow\mathbf{F}_{opt},\\ \mathbf{W}_{RF}\mathbf{W}_{BB}&\rightarrow\mathbf{W} _{opt}.\end{split}  \tag{20}\]  
Studies have shown that for the mmWave channel model, the antenna array response vectors \(\mathbf{a}_{t}(\phi_{il}^{t}),\forall i,l\) can form a finite spanning set for the row space of the channel. Moreover, when \(N_{cl}N_{ray}\leq N_{t}\), the antenna array response vectors are linearly independent with probability 1. Specifically, when \(N_{cl}N_{ray}\leq\min(N_{t},N_{t})\), \(\mathbf{a}_{t}(\phi_{il}^{t})\) can form the minimal basis for the row space of the channel [22]. Additionally, the modulus of the elements in \(\mathbf{a}_{t}(\phi_{il}^{t})\) remains constant, with only the phase varying, which perfectly matches the characteristic of analog phase shifters in antenna arrays. Therefore, the base station can select \(N_{RF}^{t}\) elements from the dictionary matrix \(\mathbf{A}_{t}=[\mathbf{a}_{t}(\phi_{1,1}^{t}),\mathbf{a}_{t}(\phi_{1,2}^{t}),\ldots,\mathbf{a}_{t}(\phi_{N_{cl},N_{ray}}^{t})]\), which consists of the transmit array response vectors, to construct the RF precoder \(\mathbf{F}_{RF}\). Subsequently, the selected array response vectors can be linearly combined through baseband precoder \(\mathbf{F}_{BB}\), ensuring that \(\mathbf{F}_{RF}\mathbf{F}_{BB}\) approximates \(\mathbf{F}_{opt}\). Similarly, on the user side, \(N_{RF}^{t}\) elements can be selected from the dictionary matrix \(\mathbf{A}_{r}=[\mathbf{a}_{r}(\phi_{1,1}^{r}),\mathbf{a}_{r}(\phi_{1,2}^{r}),\ldots,\mathbf{a}_{r}(\phi_{N_{cl},N_{ray}}^{r})]\), which consists of the receive array response vectors, to construct the RF combiner \(\mathbf{W}_{RF}\). In such a hybrid precoding system, the hybrid precoder/combiner pair \((\mathbf{F}_{RF}\mathbf{F}_{BB},\mathbf{W}_{RF}\mathbf{W}_{BB})\) corresponding to the optimal secure beam that maximizes the secrecy rate \(R_{s}\) can be obtained by performing an exhaustive search over all RF precoder/combiner pairs and selecting the pair that achieves the maximum secrecy rate. However, since both the base station and user sides utilize large-scale arrays, the dimensionality of \((\mathbf{F}_{RF},\mathbf{W}_{RF})\) is high, making it infeasible to bear the substantial computational cost and increased system processing time caused by exhaustive search. To address this issue, the OMP algorithm is proposed [22]. In this method, \(\mathbf{F}_{opt}\) is orthogonally projected onto the matrix \(\mathbf{A}_{t}\) formed by the transmit array response vectors. The algorithm quickly selects \(N_{RF}^{t}\) elements with the highest correlation to \(\mathbf{F}_{opt}\) from \(\mathbf{A}_{t}\) to construct \(\mathbf{F}_{RF}\), and then obtains \(\mathbf{F}_{BB}\) using the least squares method. Similarly, on the user side, \(\mathbf{W}_{opt}\) is orthogonally projected onto \(\mathbf{A}_{r}\), and \(N_{RF}^{r}\) elements with the highest correlation to \(\mathbf{W}_{opt}\) are selected from \(\mathbf{A}_{r}\) to construct \(\mathbf{W}_{RF}\).