## 2 System Model
### Sparsity of mmWave MIMO Channels  
The construction of mmWave channels adopts a clustered ray-based model, assuming that the channel contains \(N_{cl}\) clusters, each cluster comprising \(N_{ray}\) paths. The physical channel is determined in a nonlinear manner by the angle of departure (AoD) and angle of arrival (AoA) of each path. It can be expressed as  
\[\mathbf{H}=\sqrt{\frac{N_{t}N_{r}}{N_{cl}N_{ray}}}\sum_{i=1}^{N_{cl}}\sum_{l=1 }^{N_{ray}}\alpha_{il}\mathbf{a}_{r}(\phi_{il}^{r})\mathbf{a}_{l}^{H}(\phi_{ il}^{t}), \tag{2}\]  
Figure 1: Typical mmWave analog–digital hybrid precoding system model.  
where \(\phi_{il}^{t}\) and \(\phi_{il}^{r}\) represent the AoD and AoA for the \(l\)-th path in the \(i\)-th cluster, respectively, and \(a_{il}\) denotes the path gain. Figure 2 illustrates the propagation of scattering paths in mmWave channels.  
The mmWave MIMO channel representation in the beamspace can be expressed as  
\[\begin{array}{ll}\mathbf{H}&=\sqrt{\frac{N_{t}N_{r}}{N_{cl}N_{tr}}}\sum\limits _{i=1}^{N_{r}}\sum\limits_{j=1}^{N_{t}}\mathbf{H}_{b}(i,j)\mathbf{a}_{r}( \overline{\phi}_{i}^{r})\mathbf{a}_{l}^{H}(\overline{\phi}_{j}^{t})\\ &=\mathbf{U}_{r}\mathbf{H}_{b}\mathbf{U}_{l}^{H},\end{array} \tag{3}\]  
where \(\mathbf{H}_{b}\) is referred to as the beamspace channel matrix. \(\overline{\phi}_{i}^{r}=\arcsin\left(\frac{\lambda\overline{\phi}_{i}^{r}}{d}\right)\) and \(\overline{\phi}_{j}^{t}=\arcsin\left(\frac{\lambda\overline{\phi}_{j}^{t}}{d}\right)\) represent the AoA and AoD in the beamspace channel representation, corresponding to the uniformly spaced spatial angles \(\overline{\phi}_{i}^{r}\) and \(\overline{\phi}_{j}^{t}\), respectively. \(\lambda\) is the mmWave wavelength, and \(d\) is the antenna spacing.  
\[\begin{array}{ll}\overline{\phi}_{i}^{r}&=\frac{i-1-(N_{r}-1)/2}{N_{r}}&i \in\{1,2,\ldots,N_{r}\},\\ \overline{\phi}_{j}^{t}&=\frac{i-1-(N_{r}-1)/2}{N_{t}}&j\in\{1,2,\ldots,N_{t}\},\end{array} \tag{4}\]  
\(\mathbf{U}_{r}\) and \(\mathbf{U}_{t}\) are unitary discrete Fourier transform (DFT) matrices, whose columns are orthogonal array response vectors.  
\[\begin{array}{ll}\mathbf{U}_{r}=\frac{1}{\sqrt{N_{r}}}\left[\mathbf{a}( \overline{\phi}_{1}^{r}),\mathbf{a}(\overline{\phi}_{2}^{r}),\ldots,\mathbf{a }(\overline{\phi}_{N_{r}}^{r})\right],\\ \mathbf{U}_{t}=\frac{1}{\sqrt{N_{t}}}\left[\mathbf{a}(\overline{\phi}_{1}^{t}),\mathbf{a}(\overline{\phi}_{2}^{t}),\ldots,\mathbf{a}(\overline{\phi}_{N_{t} }^{t})\right].\end{array} \tag{5}\]  
Thus, we have  
\[\mathbf{H}_{b}=\mathbf{U}_{r}^{H}\mathbf{H}\mathbf{U}_{t}. \tag{6}\]  
The beamspace channel matrix \(\mathbf{H}_{b}\) is a unitary equivalent representation of the antenna domain channel matrix \(\mathbf{H}\), meaning that \(\mathbf{H}_{b}\) is the projection of \(\mathbf{H}\) onto the Fourier orthogonal basis. Due to the significant propagation loss and limited number of propagation paths in mmWave transmission, the channel exhibits a low-rank (sparse) characteristic. It is important to note that the channel \(\mathbf{H}\) itself is not sparse. The sparsity of the mmWave channel becomes evident in \(\mathbf{H}_{b}\).  
Figure 3a illustrates the response characteristics of the antenna domain channel \(\mathbf{H}\), which consists of six scattering clusters, each containing eight propagation paths. Figure 3b displays the response characteristics of the beamspace channel \(\mathbf{H}_{b}\). The sparsity of \(\mathbf{H}_{b}\) is clearly evident.  
Figure 2: The diagram of mmWave scattering paths.  
In fact, the sparse beamspace channel \(\mathbf{H}_{b}\) has a property that it is only related to the position of the legitimate user, reflecting the uniqueness of the channel. When an eavesdropper deviates from the legitimate user's position, he cannot obtain the beamspace channel \(\mathbf{H}_{b}\). This property can be leveraged to generate a secure beam hybrid precoder/combiner pair \((\mathbf{F}_{RF}\mathbf{F}_{BB},\mathbf{W}_{RF}\mathbf{W}_{BB})\) for the legitimate user.