## 4 Secure and Precise DM Scheme Based on A Multi-Carrier Multiple Antenna Arrays Model
### The Architecture of the LU Based on a Multi-Carrier Multiple Antenna Arrays Model  
To match the architecture of the BS, the architecture of the LU based on a multi-carrier multiple antenna arrays model is redesigned in Figure 6. In practical applications, the receive antenna array for the proposed DM scheme can be replaced by a single antenna to further simplify the DM system. After passing through the LOS channel in free space, all symbols of each symbol frame with different carriers are mixed together at the receiver side. For the recovery of the \(k\)th (\(k=1,\ldots,K\)) symbol, at first, the mixed signals should be processed with a band-pass filter with a specified center frequency \(f_{ck}\) and an appointed bandwidth \(BW\). Without loss of generality, a narrow-band channel with perfect symbol synchronization and symbol-rate sampling is assumed. Then, after the down conversion, the \(k\)th received baseband signal at time \(t\) can be expressed as  
\[y_{k}^{{}^{\prime}}(t)=\mathbf{h}^{H}(\theta_{dk})\mathbf{x}_{k}(t)\cdot e^{ -2\pi f_{ck}t}+n_{k}^{{}^{\prime}}(t), \tag{25}\]  
where \(n_{k}^{{}^{\prime}}(t)\) is the complex AWGN between the \(k\)th transmitter and the LU with the distribution being \(n_{k}^{{}^{\prime}}(t)\sim\mathcal{CN}(0,\sigma_{k}^{{}^{\prime}2})\). Substituting Equations (22)-(24) into Equation (25), and digitizing the \(k\)th received signal during the symbol period, we have  
\[y_{k} =\mathbf{h}^{H}(\theta_{dk})\mathbf{u}_{k}+n_{k}\] \[=\beta_{1}\sqrt{P_{s}}s_{k}+n_{k}, \tag{26}\]  
where the random variable \(n_{k}\) denotes the AWGN for the \(k\)th received symbol, which is i.i.d. with \(n_{k}\sim\mathcal{CN}(0,\sigma^{2})\). Figure 6: The architecture of the LU for the proposed DM based on a multi-carrier multiple antenna arrays model. Obviously, each symbol is a simple summation of the useful signal and the AWGN. Therefore, the LU can easily recover the \(k\)th confidential symbol transmitted from the \(k\)th transmitter using Equation (26). After receiving all of the transmitted symbols of an intact symbol frame during the symbol period, the received symbol frame can be written as  
\[\mathbf{y} =[y_{1},y_{2},\cdots,y_{K}]\] \[=\beta_{1}\sqrt{P_{s}}[s_{1},s_{2},\cdots,s_{K}]+\mathbf{n}\] \[=\beta_{1}\sqrt{P_{s}}\mathbf{s}+\mathbf{n}, \tag{27}\]  
where \(\mathbf{n}=[n_{1},n_{2},\cdots,n_{K}]\sim\mathcal{CN}(\mathbf{0}_{1\times K}, \sigma^{2}\mathbf{I}_{K})\) is the AWGN vector. From Equation (27), each symbol frame transmitted from the BS can be easily recovered. Similarly, for the [PERSON], the two scenarios in Section 3.2 are also considered in the following. In Case 1, let the [PERSON] be located at \(\theta_{e}\), and \(\theta_{e}\
eq\theta_{dk}\), \(k\in\{1,\cdots,K\}\). Specifically, if the [PERSON] is intended to eavesdrop the \(l\)th (\(l\in\{1,\cdots,K\}\)) symbol, its received signal at time \(t\) can be given by  
\[y_{l}^{\prime E}(t) =\mathbf{h}^{H}(\theta_{e})\mathbf{x}(t)\cdot e^{-j2\pi f_{d}t}+ n_{l}^{\prime E}(t)\] \[=\mathbf{h}^{H}(\theta_{dl})\sum_{k=1}^{K}\left[\mathbf{u}_{k}(t) \cdot e^{j2\pi f_{d}t}\right]\cdot e^{-j2\pi f_{d}t}+n_{l}^{\prime E}(t)\] \[y_{k}^{{}^{\prime}}(t)=\mathbf{h}^{H}(\theta_{dk})\mathbf{x}_{k}(t)\cdot e^{ -2\pi f_{ck}t}+n_{k}^{{}^{\prime}}(t), \tag{25}\]0 \[y_{k}^{{}^{\prime}}(t)=\mathbf{h}^{H}(\theta_{dk})\mathbf{x}_{k}(t)\cdot e^{ -2\pi f_{ck}t}+n_{k}^{{}^{\prime}}(t), \tag{25}\]1  
where \(n_{l}^{\prime E}(t)\) is the complex AWGN with the distribution \(n_{l}^{\prime E}(t)\sim\mathcal{CN}(0,\sigma_{e}^{2})\). From Equation (28), the \(l\)th received signal of the [PERSON] is composed of four components. The first is the \(l\)th distorted signal, the second is the inserted AN, the third is the equivalent mixed noise from the other transmitters, and the last is the AWGN. Furthermore, in Case 2, let the [PERSON] be located at \(\theta_{e}\), and \(\theta_{e}=\theta_{dl}\), \(l\in\{1,\cdots,K\}\).