## 3 SM and MP-WFRFT Aided Scheme Based On FDA
### The Radiating Signal Processed by SM and MP-WFRFT  
The architecture of transmit station is shown in Figure 3. In this paper, we innovatively apply two key modulation techniques into FDA-DM. Before transmitting the confidential signal to LUs, we first use the SM module for the confidential signal stream \(x(t)\) and \(x(t)\) is normalized, i.e., \(\mathbb{E}[|x(t)|^{2}]=1\). Then, we divide the confidential information bit stream into blocks. As mentioned in previous section, each block contains \(\log_{2}(KM^{Q})\) bits. The SM is operated to the confidential information block, which yields the transmitting symbol vector  
\[\mathbf{u}(t)=\mathcal{U}[x(t)]=[u_{1}(t),u_{2}(t),\ldots,u_{K}(t)]^{T}, \tag{8}\]  
where \(u_{k}(t)\) is the transmitting symbol for the \(k\)-th LU, and \(\mathcal{U}[\cdot]\) is the SM mapping.  
Second, the transmitting symbol vector \(\mathbf{u}(t)\) is performed by MP-WFRFT with parameters \([\alpha_{s},\mathbf{m}_{s},\mathbf{c}_{s}]^{T}\), which can be expressed as  
\[\mathbf{v}=\bar{\mathfrak{F}}_{\alpha_{s}}^{(\mathbf{m}_{s},\mathbf{c}_{s})}( \mathbf{u})=w_{0}\mathbf{u}+w_{1}\mathcal{F}(\mathbf{u})+w_{2}\mathbf{Pu}+w_{ 3}\mathbf{P}\mathcal{F}(\mathbf{u}), \tag{9}\]  
Figure 2: System model.  
The vector \(\mathbf{u}(t)\) is rotated in the time-frequency plane to realize the time-frequency redistribution of signal power. Therefore, only when the users rotates the signal in same angle but the opposite direction can the signal power be concentrated.  
Before radiating, we design the beamforming matrix \(\mathbf{D}\) to further process the symbol vector \(\mathbf{v}\) to match all transmit antennas. The beamforming matrix \(\mathbf{D}\) is given by  
\[\mathbf{D}=[\mathbf{d}_{1}(t,\mathbf{f}),\mathbf{d}_{2}(t,\mathbf{f}),\ldots,\mathbf{d}_{K}(t,\mathbf{f})], \tag{10}\]  
where \(\mathbf{d}_{k}(t,\mathbf{f})=[d_{k,1}(t,\mathbf{f}),d_{k,2}(t,\mathbf{f}), \ldots,d_{k,N}(t,\mathbf{f})]^{T}\), for \(k=1,2\ldots,K\), is the beamforming vector to process symbol \(v_{k}(t)\) that is transmitted to \(k\)-th LU.  
In order to obtain the beamforming matrix \(\mathbf{D}\), the locations of LUs are assumed to be previously known by transmit station, i.e., the steering matrix \(\mathbf{H}_{L}(t)\) is known in advance. Next, we design the beamforming matrix \(\mathbf{D}\) based on the rules that (1) the intended LU effectively receives the corresponding confidential messages, while the undesired LUs cannot obtain the messages, and (2) the transmit power is minimum, while satisfying communication performance requirements of each LU. Therefore, the beamforming matrix \(\mathbf{D}\) is designed by  
\[\mathbf{H}_{L}^{H}(t)\mathbf{D}=\mathbf{A}\mathbf{I}_{\mathbf{K}}, \tag{11}\]  
where \(\mathbf{A}=\text{diag}[\xi]\), in which \(\boldsymbol{\zeta}\overset{\Delta}{=}[\sqrt{\xi_{1}},\sqrt{\xi_{2}},\ldots, \sqrt{\xi_{k}},\ldots,\sqrt{\xi_{k}}]\), and \(\xi_{k}\) is the minimum desired power, for \(k=1,2,\cdots,K\).  
According to the pseudo-inverse concept, we obtain \(\mathbf{D}\) as  
\[\mathbf{D}=\mathbf{H}_{L}(t)(\mathbf{A}^{-1})^{H}[\mathbf{A}^{-1}\mathbf{H}_ {L}^{H}(t)\mathbf{H}_{L}(t)(\mathbf{A}^{-1})^{H}]^{-1}, \tag{12}\]  
After processed by the beamforming matrix, the radiating signal for the \(N\) antennas is given by  
\[\mathbf{s}=\mathbf{D}\mathbf{v}=\mathbf{D}\mathbf{s}_{\mathbf{a}_{\mathbf{s}} }^{\{\mathbf{m}_{\mathbf{s}},\mathbf{c}_{\mathbf{s}}\}}[\mathcal{U}(\mathbf{ x})], \tag{13}\]