## 2 System Model and Problem Formulation
Since the HP-SN utilizes PS scheme to simultaneously implement ID and EH, the received signal at the HP-SN\({}_{k}\) is divided into ID portion and EH portion by the PS ratio \(\rho_{k}\in(0,1]\). Hence, the split signal for ID part and EH part at the HP-SN\({}_{k}\) can be expressed as, respectively,  
\[y_{c,k}^{ID} =\sqrt{\rho_{k}}\big{(}\mathbf{h}_{c,k}^{H}\mathbf{w}_{k}s_{s,k}+ \mathbf{h}_{c,k}^{H}\mathbf{z}+\mathbf{h}_{c,k}^{H}\sum_{i=1,i\
eq k}^{K}( \mathbf{w}_{i}s_{s,i})+\mathbf{g}_{c,k}^{H}\sum_{m=1}^{M}\mathbf{v}_{m}s_{m}+n _{a,k}\big{)}+n_{p,k},\] \[y_{c,k}^{EH} =\sqrt{1-\rho_{k}}\big{(}\mathbf{h}_{c,k}^{H}\mathbf{w}_{k}s_{s,k} +\mathbf{h}_{c,k}^{H}\mathbf{z}+\mathbf{h}_{c,k}^{H}\sum_{i=1,i\
eq k}^{K}( \mathbf{w}_{i}s_{s,i})+\mathbf{g}_{c,k}^{H}\sum_{m=1}^{M}\mathbf{v}_{m}s_{m}+n _{a,k}\big{)},\]  
where \(n_{p,k}\sim\mathcal{CN}(0,\delta_{p,k}^{2})\) is the additional Gaussian noise resulted in the RF-to-signal conversion for the ID part at the HP-SN\({}_{k}\). Thus, the channel capacity of the HP-SN\({}_{k}\) is expressed as  
\[R_{c,k}=\log\bigg{(}1+\frac{\rho_{k}\mathbf{h}_{c,k}^{H}\mathbf{w}_{k}\mathbf{ w}_{k}^{H}\mathbf{h}_{c,k}}{\rho_{k}\big{(}\mathbf{h}_{c,k}^{H}(\sum_{j\
eq k} \mathbf{w}_{j}\mathbf{w}_{j}^{H}+\mathbf{z}\mathbf{z}^{H})\mathbf{h}_{c,k}+ \mathbf{g}_{c,k}^{H}\sum_{m=1}^{M}(\mathbf{v}_{m}\mathbf{v}_{m}^{H})\mathbf{g}_{ c,k}+\sigma_{a,k}^{2})+\delta_{p,k}^{2}}\bigg{)}. \tag{1}\]  
The channel capacity of the LP-SN\({}_{l}\) for decoding the desired signal of the HP-SN\({}_{k}\) can be represented as  
\[R_{c,lk}=\log\bigg{|}\mathbf{I}+\bigg{(}\sigma_{e,l}^{2}\mathbf{I}+\big{(} \mathbf{H}_{e,l}^{H}\big{(}\sum_{j\
eq k}\mathbf{w}_{j}\mathbf{w}_{j}^{H}+ \mathbf{z}\mathbf{z}^{H}\big{)}\mathbf{H}_{e,l}+\mathbf{G}_{e,l}^{H}\sum_{m=1}^ {M}(\mathbf{v}_{m}\mathbf{v}_{m}^{H})\mathbf{G}_{e,l}\big{)}\bigg{)}^{-1} \mathbf{H}_{e,l}^{H}\mathbf{w}_{k}\mathbf{w}_{k}^{H}\mathbf{H}_{e,l}\bigg{|}. \tag{2}\]  
Thus, the secrecy capacity of the HP-SN\({}_{k}\) can be written as  
\[C_{c,k}=\bigg{[}R_{c,k}-\max_{l}R_{c,lk}\bigg{]}^{+},\ \forall k,l. \tag{3}\]Additionally, the channel capacity of the M-SN\({}_{m}\) can be expressed as  
Figure 2:0  
Suppose that the LP-SN\({}_{l}\) is an eavesdropper to decode the message for the M-SN\({}_{m}\). The channel capacity of the LP-SN\({}_{l}\) for decoding the M-SN\({}_{m}\) is given as  
Figure 2:1  
Hence, following the concept of secrecy capacity in [20; 21; 22], the secrecy capacity of the M-SN\({}_{m}\) can be written as  
Figure 2:2  
where \([a]^{+}\) indicates the \(max(0,a)\). Moreover, the HP-SNs employs the PS ratio to harvest the power carried from the RF signal. In order to harvest the carried energy, the LP-SN receiver does not need to transform the received signal to the baseband. However, due to the law of energy conservation, we assume that the total harvested energy at the HP-SN\({}_{k}\) and the LP-SN\({}_{l}\), i.e., the RF-band power normalized by the baseband symbol, denoted by \(E_{c,k}\) and \(E_{e,l}\), respectively, from all \(N_{E}\) receive antennas at the LP-SN receiver is proportional to that of the received RF baseband signal, i.e.,  
Figure 2:3 Figure 2:4  
where \(0\leq\eta_{c,k}\leq 1\) and \(0\leq\eta_{e,l}\leq 1\) denote the energy conversion efficiency at the HP-SN\({}_{k}\) and LP-SN\({}_{l}\), respectively. In this paper, our objective is to secure the WIT to legitimated users (i.e., M-SNs and HP-SNs) in the presence of potential eavesdroppers (i.e., LP-SNs) by jointly designing the AN-aided transmit beamforming. Moverover, we aim to promote the MSC performance by guaranteeing all M-SNs can achieve a certain secrecy capacity level. To this end, we maximize the MSC of the M-SN\({}_{m}\) under the secrecy capacity constraint at the HP-SNs, the harvested energy constraint at the HP-SNs and LP-SNs, as well as the transmit power constraint at the macrocell sink node and femtocell sink node.