# Deep Learning-Based Secure MIMO Communications with Imperfect CSI for Heterogeneous Networks
## 3 System Model  
Figure 1 depicts the model of secure MIMO communications for heterogeneous networks. In the considered system, there is a macro BS, a femto BS and a terminal user. It is assumed that three types of nodes are all equipped with multiple antennas, and the numbers of antennas are denoted as \(N_{M}\), \(N_{F}\) and \(N_{U}\), respectively. In the heterogeneous networks, macro BS and femto BS work cooperatively to provide the wireless coverage. Specifically, the users located in the hot spots area are associated with the macro BS, and the users located in the network edge are served by the femto BS. We use \(\mathbf{H}_{FM}\) and \(\mathbf{H}_{UM}\) to denote the wireless channel fading matrix from the macro BS to the femto BS and the users, respectively. In addition, the channel fading matrix is modeled as time-varying and flat fading using the classical Jakes model [37]. Thus, the correlation coefficient between adjacent samples is given as  
\[\rho=J_{0}(2\pi f_{d}), \tag{1}\]  
where \(f_{d}\) is the normalized doppler frequency spread and \(J_{0}(\cdot)\) is the zero-order Bessel function of the first kind. Thus, the channel fading matrix can be calculated as  
\[\mathbf{H}_{UIM}(n)=\rho\mathbf{H}_{UIM}(n-1)+\sqrt{1-\rho^{2}}\mathbf{N}_{UIM}(n), \tag{2}\]  
where \(n\) denotes the sample time and \(N_{UIM}(n)\) denotes the additional white noise matrix with the same size of \(H_{UIM}(n)\). The same equation can be applied on \(\mathbf{H}_{FM}\) as follows:  
\[\mathbf{H}_{FM}(n)=\rho\mathbf{H}_{FM}(n-1)+\sqrt{1-\rho^{2}}\mathbf{N}_{FM}(n). \tag{3}\]  
Please note that in the following sections, the sample time \(n\) is omitted without loss of generality. It is well-known that the deep learning networks can effectively capture the correlation features of the training data set. Spatial correlation or antenna correlation, which may achieve dimensionality reduction, is an important challenge for MIMO systems [38; 39]. Also, it will be a direction of our future work. Since the time correlated MIMO channel model is adopted in this paper, we can use DCNN to obtain more accurate CSI from the outdated CSI. Actually, considering the phase rotation introduced by the channel matrix, the outdated CSI is necessary to assist the data recovery at the user. Due to the open nature of the heterogeneous networks, the femto BS may intercept the signal transmitted from the macro BS to the users. To prevent the information leakage to the femto BS, the macro BS can zero-forcing the equivalent channel matrix of the femto BS by the null-space technique. In this case, the macro BS first obtains the CSI \(\mathbf{H}_{FM}\) between the macro BS and the femto BS. Then the null-space eigenvectors can be produced by applying eigenvalue decomposition on the autocorrelation matrix, i.e., \(\mathbf{H}_{FM}^{*}\mathbf{H}_{FM}\), that is  
\[(\mathbf{v},\mathbf{V})=Eig(\mathbf{H}_{FM}^{*}\mathbf{H}_{FM}), \tag{4}\]  
Figure 1: Secure MIMO communications for heterogeneous networks. where \(Eig(\cdot)\) denotes the eigenvalue decomposition, \(\mathbf{v}\) denotes the eigenvalues in ascending order and \(\mathbf{V}\) are the corresponding eigenvectors. Considering the size limitation for both the femto BS and users, it is reasonable to assume that the antenna number of the macro BS is larger than the femto BS and users, i.e., \(N_{M}>N_{F},N_{M}>N_{U}\). Thus, the number of zero eigenvalues can be given as  
\[N_{D}=N_{M}-N_{F}. \tag{5}\]  
Please note that \(N_{D}\) is also the number of null-space vectors for \(\mathbf{H}_{FM}\), which is given as  
\[\mathbf{B}=\mathbf{V}_{:N_{D}} \tag{6}\]  
where \(\mathbf{V}_{:N_{D}}\) denotes the first \(N_{D}\) column vectors of \(\mathbf{V}\). Thus, the beamforming matrix \(\mathbf{B}\), which is used by the macro BS to transmit messages to its associated user, lies in the null-space of \(\mathbf{H}_{FM}\).