## 3 Secure and Precise DM Scheme Based on A Single-Carrier Multiple Antenna Arrays Model
### The Architecture of the LU Based on a Single-Carrier Multiple Antenna Arrays Model  
The architecture of the LU based on a single-carrier multiple antenna arrays model is demonstrated in Figure 4. To fairly compare the proposed DM technique and other DM techniques, without loss of generality, the normalized narrow-band LOS AWGN channel with perfect symbol-rate sampling and synchronization in free space is assumed throughout the paper. After down-converting to baseband using the same carrier \(f_{c}\), the received signal \(\mathbf{y}^{{}^{\prime}}=[y^{{}^{\prime}}_{1},\cdots,y^{{}^{\prime}}_{N}]^{T}\) of the LU at time \(t\) can be written as  
\[\mathbf{y}^{{}^{\prime}}(t)=e^{-j2\pi f_{c}t}\sum_{k=1}^{K}\mathbf{x}_{k}(t)+ \mathbf{n}^{{}^{\prime}}, \tag{10}\]  
where \(\mathbf{n}^{{}^{\prime}}=[n^{{}^{\prime}}_{1},\cdots,n^{{}^{\prime}}_{N}]^{T} \sim\mathcal{CN}(\mathbf{0}_{N\times 1},\sigma^{2}_{\mathbf{n}^{{}^{\prime}}} \mathbf{I}_{N})\) is the complex AWGN vector. Substituting Equations (3), (4) and (8) into Equation (10), the received signal \(\mathbf{y}^{{}^{\prime}}\) can be further simplified as  
\[\mathbf{y}^{{}^{\prime}}=\sum_{k=1}^{K}\left(\beta_{1}\sqrt{P_{s}}\mathbf{h}( \theta_{dk})s_{k}+\beta_{2}\sqrt{P_{s}}\mathbf{T}_{k}\mathbf{z}_{k}\right)+ \mathbf{n}^{{}^{\prime}}. \tag{11}\]  
As we can see from Equation (11), the \(K\) symbols in each symbol frame are mixed together at the receiver side. We have to design a receive multi-beamforming scheme to recover the \(K\) symbols. Each set of weight coefficients corresponds to a receive beam for the recovery of one symbol. For example, to obtain the \(l\)th (\(l\in\{1,2,\cdots,K\}\)) symbol \(s_{l}\) from the direction \(\theta_{dl}\), it is necessary to steer the array nulls precisely to the other \(K-1\) directions. The subspace of the \(K-1\) interference directions is given by  
\[\mathbf{A}=[\mathbf{h}(\theta_{dl}),\cdots,\mathbf{h}(\theta_{d(l-1)}), \mathbf{h}(\theta_{d(l+1)}),\cdots,\mathbf{h}(\theta_{dK})]. \tag{12}\]  
The orthogonal complement space of \(\mathbf{A}\) is denoted by \(\mathbf{P}^{\perp}_{\mathbf{A}}\), which is given by  
\[\mathbf{P}^{\perp}_{\mathbf{A}}=\mathbf{I}_{N}-\mathbf{A}(\mathbf{A}^{H} \mathbf{A})^{-1}\mathbf{A}^{H}. \tag{13}\]  
Obviously, \(\mathbf{P}^{\perp}_{\mathbf{A}}\) is a Hermitian matrix. We have  
\[\mathbf{P}^{\perp}_{\mathbf{A}}=(\mathbf{P}^{\perp}_{\mathbf{A}})^{H}. \tag{14}\]  
Therefore, the weight vector of the \(l\)th receive beam can be designed as  
\[\mathbf{w}_{l}=\mathbf{P}^{\perp}_{\mathbf{A}}\mathbf{h}(\theta_{dl}). \tag{15}\]  
As is well known from matrix theory, if subspace \(\mathbf{A}\) is orthogonal to subspace \(\mathbf{B}\), then the vector projected in subspace \(\mathbf{A}\) is also perpendicular to subspace \(\mathbf{B}\). Hence, arbitrary vector \(\mathbf{h}(\theta_{dk})\), \(k=\{1,\cdots,l-1,l+1,\cdots,K\}\) will satisfy that  
\[\mathbf{w}^{H}_{l}\mathbf{h}(\theta_{dk})=0. \tag{16}\]  
Figure 4: The architecture of the LU for the proposed DM based on a single-carrier multiple antenna arrays. In other words, the beam formed by the weight vector \(\mathbf{w}_{l}\) can result in null steering in other \(K-1\) desired directions. Let \(\xi=\mathbf{h}^{H}(\theta_{dl})\mathbf{A}(\mathbf{A}^{H}\mathbf{A})^{-1}\mathbf{A }^{H}\mathbf{h}(\theta_{dl})\) and \(n_{l}=\mathbf{w}_{l}^{H}\mathbf{n}^{{}^{\prime}}\). Therefore, the \(l\)th (\(l\in\{1,2,\cdots,K\}\)) received symbol \(y_{l}\) after the receive multi-beamforming processing can be expressed as  
\[y_{l}=\mathbf{w}_{l}^{H}\mathbf{y}^{{}^{\prime}}\!\!=\!\!s_{l}\beta_{1}\sqrt{P_ {s}}(1-\xi)+n_{l}, \tag{17}\]  
where \(\xi\) is a tiny real-valued scaling factor and \(n_{l}\) is the AWGN with zero mean and variance \(\sigma_{n_{l}}^{2}\). Proof. : Please refer to Appendix A. It is noted that the received symbol \(y_{l}\) is a simple summation of the useful signal \(s_{l}\) and AWGN \(n_{l}\). From Equation (17), the LU can easily recover the \(l\)th (\(l\in\{1,2,\cdots,K\}\)) symbol transmitted from the BS.