## 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
Finally, we can represent the \(K\) received symbols as a received vector, i.e., a received symbol frame,  
\[\mathbf{y}^{{}^{\prime}}(t)=e^{-j2\pi f_{c}t}\sum_{k=1}^{K}\mathbf{x}_{k}(t)+ \mathbf{n}^{{}^{\prime}}, \tag{10}\]0  
However, for the Eve, we consider two scenarios. One case is that the Eve is not in the same direction as the LU for any antenna arrays. The other worse case is that the Eve is in the same direction as the desired receiver for one antenna array and not in the same direction for another antenna array. In Case 1, the assumption condition is denoted as \(\theta_{dl}\
eq\theta_{ek}\) for arbitrary \(l,k\in\{1,2,\cdots,K\}\). The received message can be written as Equation (19), where \(n_{l}^{E}=\mathbf{h}^{H}(\theta_{el})\mathbf{n}^{{}^{\prime}}\sim\mathcal{CN} (0,\sigma_{n_{l}^{E}}^{2})\) is the AWGN with zero mean and variance \(\sigma_{n_{l}^{E}}^{2}\). \[\mathbf{y}^{{}^{\prime}}(t)=e^{-j2\pi f_{c}t}\sum_{k=1}^{K}\mathbf{x}_{k}(t)+ \mathbf{n}^{{}^{\prime}}, \tag{10}\]1 \[\mathbf{y}^{{}^{\prime}}(t)=e^{-j2\pi f_{c}t}\sum_{k=1}^{K}\mathbf{x}_{k}(t)+ \mathbf{n}^{{}^{\prime}}, \tag{10}\]2 \[\mathbf{y}^{{}^{\prime}}(t)=e^{-j2\pi f_{c}t}\sum_{k=1}^{K}\mathbf{x}_{k}(t)+ \mathbf{n}^{{}^{\prime}}, \tag{10}\]3  
It can be seen from Equation (19) that the received signal is made up of four components. The first is the scrambled signal. When \(\theta_{dl}\
eq\theta_{ek}\), we have \(\mathbf{h}^{H}(\theta_{el})\mathbf{h}(\theta_{dl})=\delta\cdot e^{i\Delta\varphi}\). It is obvious that there exists an amplitude difference \(\delta\beta_{1}\sqrt{P_{s}}\) and a phase difference \(\Delta\varphi\) between the received symbol and the original signal \(s_{l}\) for the first part, which can disrupt the standard constellation format. The second component is the equivalent AN caused by the other \(K-1\) symbols in each symbol frame, which results in a high interference level for the recovery of symbol \(s_{l}\). The third part is the superimposed AN from the \(K\) transmit antenna arrays. The fourth component is the AWGN. Therefore, the Eve can hardly recover the confidential information from the seriously interfered signals, which indicates that the transmission security can be guaranteed in the first scenario. In Case 2, a worse case, the assumption condition can be described as \(\theta_{el}\!\!=\!\!\theta_{dl}\), for a fixed \(l\), (\(l\in\{1,2,\cdots,K\}\)), and \(\theta_{el}\
eq\theta_{dk}\), for arbitrary \(k,k\in\{1,\cdots,l-1,l+1,\cdots,K\}\). Then, the received signal from the \(l\)th (\(l\in\{1,2,\cdots,K\}\)) transmit antenna array can be expressed as Equation (20). \[\mathbf{y}^{{}^{\prime}}(t)=e^{-j2\pi f_{c}t}\sum_{k=1}^{K}\mathbf{x}_{k}(t)+ \mathbf{n}^{{}^{\prime}}, \tag{10}\]4 \[\mathbf{y}^{{}^{\prime}}(t)=e^{-j2\pi f_{c}t}\sum_{k=1}^{K}\mathbf{x}_{k}(t)+ \mathbf{n}^{{}^{\prime}}, \tag{10}\]5Likewise, the other received signal from the \(k\)th (\(k\in\{1,\cdots,l-1,l+1,\cdots,K\}\)) transmit antenna array can be expressed as Equation (21), where \(n_{k}=\mathbf{h}^{H}(\theta_{ck})\mathbf{n}^{{}^{\prime}}\) is the AWGN with PDF being \(\mathcal{CN}(0,\sigma_{n_{k}^{2}}^{2})\). \[\mathbf{y}^{{}^{\prime}}(t)=e^{-j2\pi f_{c}t}\sum_{k=1}^{K}\mathbf{x}_{k}(t)+ \mathbf{n}^{{}^{\prime}}, \tag{10}\]6 \[\mathbf{y}^{{}^{\prime}}(t)=e^{-j2\pi f_{c}t}\sum_{k=1}^{K}\mathbf{x}_{k}(t)+ \mathbf{n}^{{}^{\prime}}, \tag{10}\]7 \[\mathbf{y}^{{}^{\prime}}(t)=e^{-j2\pi f_{c}t}\sum_{k=1}^{K}\mathbf{x}_{k}(t)+ \mathbf{n}^{{}^{\prime}}, \tag{10}\]8  
It can also be observed from Equation (20) that the received signal from the \(l\)th (\(l\in\{1,2,\cdots,K\}\)) antenna array is severely contaminated by the other \(K-1\) data streams, \(K-1\) inserted AN and the AWGN. Similarly, from Equation (21), we can find that the received signal from the \(k\)th (\(k\in\{1,\cdots,l-1,l+1,\cdots,K\}\)) antenna array is also scrambled by the term \(\mathbf{h}^{H}(\theta_{ck})\mathbf{h}(\theta_{dk})\), contaminated by the equivalent AN, inserted AN, and the AWGN.