## 3 Secure and Precise DM Scheme Based on A Single-Carrier Multiple Antenna Arrays Model
### The Architecture of BS Based on a Single-Carrier Multiple Antenna Arrays Model  
The architecture of the BS is shown in Figure 3, which employs \(K\) ULAs with all antenna arrays emitting the same carrier to steer the \(K\) confidential information beams towards the LU simultaneously. Unlike traditional beamforming, the AN-aided beamforming technique has been adopted extensively under the background of secure wireless communications due to its perfect security performance. The received information in the desired direction will not be affected by AN, while the messages in other directions will be seriously interfered, which is beneficial for PLS enhancement. Therefore, AN-aided beamforming technique is introduced to achieve DM in the proposed scheme to achieve secure and precise wireless transmission. Let \(\mathbf{s}=[s_{1},s_{2},\cdots,s_{K}]\) denote the symbol vector formed by combining the \(K\) pre-transmitting baseband symbols via multiple antenna arrays. From another perspective, \(\mathbf{s}\) can also be regarded as a \(K\)-element symbol frame. The \(k\)th symbol \(s_{k}\) is valued from the set \(\mathbb{B}=\{b_{1},b_{2},\cdots,b_{M}\}\) with \(card(\mathbb{B})=M\), which comprises all symbols belonging to an \(M\)-ary amplitude and phase modulation, such as PSK (phase shift keying) or QAM (quadrature amplitude modulation), i.e., \(s_{k}\in\mathbb{B}\). Suppose each baseband symbol is normalized, i.e., satisfies \(\mathbb{E}[|s_{k}^{*}s_{k}|]=1\). At first, do serial-to-parallel conversion for each symbol vector \(\mathbf{s}\) to match \(K\) transmit antenna arrays. Next, before transmitting, the \(k\)th symbol \(s_{k}\) should be processed by AN-aided beamforming to match the \(N\) elements of the \(k\)th transmit antenna array. The transmitted signal vector \(\mathbf{u}_{k}=[u_{1k},u_{2k},\cdots,u_{Nk}]^{T}\) for the \(N\) antenna elements can be given by  
\[\mathbf{u}_{k}=\beta_{1}\sqrt{P_{s}}\mathbf{v}_{k}s_{k}+\beta_{2}\sqrt{P_{s} }\mathbf{T}_{k}\mathbf{z}_{k}, \tag{3}\]  
where \(P_{s}\) is the total transmit power for each symbol; \(\mathbf{z}_{k}\) is an \(N\)-tuple vector consisting of independent and identically distributed (i.i.d.) complex Gaussian random variables with the probability density  
Figure 3: The architecture of the BS for the proposed DM scheme based on a single-carrier multiple antenna arrays model. function (PDF) being \(\mathcal{CN}(\mathbf{0}_{N\times 1},\mathbf{I}_{N})\), i.e., the inserted AN for the \(k\)th antenna array; \(\beta_{1}\in(0,1]\) and \(\beta_{2}\in[0,1)\) are information and AN power allocation factors, respectively, satisfying \(\beta_{1}^{2}+\beta_{2}^{2}=1\); \(\mathbf{v}_{k}\in\mathbb{C}^{N\times 1}\) is the transmit beamforming vector, which projects the useful signal \(s_{k}\) to the desired direction; and \(\mathbf{T}_{k}\in\mathbb{C}^{N\times N}\) is the projection matrix for controlling the direction of AN. Then, our primary task is to find the transmit beamforming vector \(\mathbf{v}_{k}\) and the AN projection matrix \(\mathbf{T}_{k}\) for the \(k\)th transmit antenna array. Here, the AN projection matrix \(\mathbf{T}_{k}\) enforces the AN into null-space of the desired steering vector or the eavesdropper steering vector with a slight or no leakage of AN power to the desired steering space, while the precoding vector for the useful information is to convey the major power of useful information to the desired steering space with a small fraction or no leakage to its null-space. Different from traditional beamforming, here, DM signals are synthesized in the baseband. The excitation signal vector of the useful signal corresponds to the steering vector of the desired direction.