## 3 SM and MP-WFRFT Aided Scheme Based On FDA
### The Received Signal of LUs and Eves  
The receiving structure with cooperative LUs is shown in Figure 4. Because of cooperative LUs, we combine all LUs received signals together as an LUs received vector, i.e., \(\mathbf{y}_{L}(t)=[y_{L_{1}}(t),y_{L_{2}}(t),\ldots,y_{L_{K}}(t)]^{T}\). In this paper, assuming that (1) the synchronization of time and frequency  
Figure 3: The architecture of the transmit station. is perfect in the ideal scenario, and (2) the MP-WFRFT parameters are securely shared between the transmit station and LUs. As shown in Figure 4, the received symbol is first operated by inverse MP-WFRFT with \((\alpha_{l},\mathbf{m}_{l},\mathbf{c}_{l})\), which yields  
\[\mathbf{y}_{L}(t)=\mathfrak{F}_{\bar{a}_{l}}^{(\mathbf{m}_{l},\mathbf{c}_{l})}( \mathbf{y}_{L}^{{}^{\prime}}(t))=\mathfrak{F}_{\bar{a}_{l}}^{(\mathbf{m}_{l}, \mathbf{c}_{l})}(\mathbf{H}_{L}^{H}(t)\mathbf{s}(t))+\mathfrak{F}_{\bar{a}_{l}}^{( \mathbf{m}_{l},\mathbf{c}_{l})}(\mathbf{n}_{L}(t)) \tag{14}\]  
where the shared parameters \((\alpha_{l},\mathbf{m}_{l},\mathbf{c}_{l})=(-\alpha_{s},\mathbf{m}_{s},\mathbf{c}_{s})\). Based on (9), (11), and (13), (14) can be further simplified as  
\[\mathbf{y}_{L}(t) =\mathfrak{F}_{-\bar{a}_{s}}^{(\mathbf{m}_{s},\mathbf{c}_{s})}( \mathbf{A}\mathbf{v}(t))+\mathfrak{F}_{-\bar{a}_{s}}^{(\mathbf{m}_{s},\mathbf{c} _{s})}(\mathbf{n}_{L}(t))=\mathbf{A}\mathfrak{F}_{-\bar{a}_{s}}^{(\mathbf{m}_ {s},\mathbf{c}_{s})}\mathfrak{F}_{\bar{a}_{s}}^{(\mathbf{m}_{s},\mathbf{c}_{s})}( \mathbf{u}(t))+\mathfrak{F}_{-\bar{a}_{s}}^{(\mathbf{m}_{s},\mathbf{c}_{s})}( \mathbf{n}_{L}(t))\] \[=\mathbf{A}\mathbf{u}(t)+\mathbf{n}_{L}^{{}^{\prime}}(t) \tag{15}\]  
where \(\mathbf{n}_{L}(t)=[u_{L_{1}}(t),n_{L_{2}}(t),\ldots,n_{L_{k}}(t),\ldots,n_{L_{ k}}(t)]^{T}\) is the AWGN vector with each element having zero mean and variance \(\sigma_{L_{k}}^{2}\), i.e., \(\mathbf{n}_{L}(t)\sim\mathcal{CN}(\mathbf{0}_{K\times 1},\sigma_{L}^{2} \mathbf{I}_{K})\). \(\mathbf{n}_{L}^{{}^{\prime}}(t)=[n_{L_{1}}^{{}^{\prime}}(t),n_{L_{2}}^{{}^{ \prime}}(t),\ldots,n_{L_{k}}^{{}^{\prime}}(t),\ldots,n_{L_{k}}^{{}^{\prime}}(t )]^{T}\) is the AWGN vector after MP-WFRFT, which remains the same distribution characteristics, i.e., \(\mathbf{n}_{L}^{{}^{\prime}}(t)\sim\mathcal{CN}(\mathbf{0}_{K\times 1}, \sigma_{L}^{2}\mathbf{I}_{K})\). Thereinto, the received symbol of \(k\)-th LU is given by  
\[y_{L_{k}}(t) =\mathfrak{F}_{-\bar{a}_{s}}^{(\mathbf{m}_{s},\mathbf{c}_{s})}( \mathbf{h}_{L_{k}}^{H}(t)\mathbf{s}(t))+n_{L_{k}}^{{}^{\prime}}(t)=\mathbf{h} _{L_{k}}^{H}(t)\mathbf{d}_{k}(t)u_{k}(t)+\mathbf{h}_{L_{k}}^{H}(t)\sum_{i=1,j \
eq k}^{K}\mathbf{d}_{i}(t)u_{i}+n_{L_{k}}^{{}^{\prime}}(t)\] \[=\sqrt{\zeta_{k}}u_{k}(t)+n_{L_{k}}^{{}^{\prime}}(t) \tag{16}\]  
From (15) and (16), it can be seen that each LU can effectively receive the corresponding transmitting symbol under the control of desired received power, and the transmitting symbol vector \(\mathbf{u}(t)\) is recovered via inverse MP-WFRFT with cooperative LUs. Then, after the correct reception and inverse MP-WFRFT of all LUs, the confidential signal stream x(t) is obtained by demapping the transmitting symbol vector \(\mathbf{u}(t)\). Next, we assume there are \(J\) passive Eves located in different positions intercepting the confidential information. We define \((r_{ej},\theta_{ej})\) as the coordinates of Eve \(j\) and use the steering matrix \(\mathbf{H}_{E}(t)\) to denote steering vectors of all Eves, i.e.,  
\[\mathbf{H}_{E}(t)\overset{\Delta}{=}[\mathbf{h}_{E_{1}}(t),\mathbf{h}_{E_{2}} (t),\cdots,\mathbf{h}_{E_{j}}(t),\cdots,\mathbf{h}_{E_{j}}(t)], \tag{17}\]  
where \(\mathbf{h}_{E_{j}}(t)\) is the instantaneous normalized steering vector of \(j\)-th Eve. Furthermore, we consider a worse case in which Eves in different positions can cooperate with each other.