## 3 Power Allocation of SCO\(-\)AN
### Objective Function of Power Allocation
#### 3.1.2 Objective Function of Power Allocation for MIMO Communication System  
For the MIMO communication system, the power for each transmission is \(P_{0,m}\). Therefore,  
\[\sigma_{g}^{2}+\sigma_{u}^{2}\leq P_{0,m}. \tag{34}\]  
The initial secrecy capacity is \(C_{sdc}^{0,m}\). For the perfect channel estimation, the initial secrecy capacity is given by:  
\[\log\left|\mathbf{I}\sigma_{n}^{2}+\mathbf{H}_{k}\mathbf{Z}_{k}\mathbf{Z}_{k} ^{\dagger}\mathbf{H}_{k}^{\dagger}\mathbf{x}_{0}\right|-\log\left(\frac{| \mathbf{G}_{k}\mathbf{Z}_{k}\mathbf{Z}_{k}^{\dagger}\mathbf{G}_{k}^{\dagger }\mathbf{y}_{0}+\mathbf{I}\sigma_{g}^{2}\mathbf{Z}_{k}\mathbf{Z}_{k}^{ \dagger}\mathbf{G}_{k}^{\dagger}\mathbf{x}_{0}|}{|\mathbf{G}_{k}\mathbf{Z}_{ k}\mathbf{Z}_{k}^{\dagger}\mathbf{G}_{k}^{\dagger}\mathbf{y}_{0}+\mathbf{I}\sigma_{g}^{2}|} \right)=C_{sdc}^{0,m}. \tag{35}\]  
The Lagrange function of (40) is:  
\[L(x,y,\mu,\lambda)=f(x,y)-\mu_{1}h_{1}(x,y)-\sum_{j=1,2,3,4}\lambda_{j}g_{j}(x,y), \tag{41}\]  
where  
\[ \begin{split} f(x,y)&=-\log\left(\frac{\left|\mathbf{ K}_{H}+(\mathbf{H}_{\alpha}+\mathbf{\hat{H}})\mathbf{Z}_{2}\mathbf{Z}_{2}^{ \dagger}(\mathbf{H}_{\alpha\alpha}+\mathbf{\hat{H}})^{\dagger}x\right|}{ \left|\mathbf{K}_{H}\right|}\right)+\log\left(\frac{\left|\mathbf{K}_{G}+ \mathbf{G}_{2}\mathbf{Z}_{2}\mathbf{Z}_{2}^{\dagger}\mathbf{G}_{2}^{\dagger} \mathbf{x}_{2}^{\dagger}\right|}{\left|\mathbf{K}_{G}\right|}\right)\\ g_{1}(x,y)&=\log\left(\frac{\left|\mathbf{K}_{H}+( \mathbf{H}_{\alpha\alpha}+\mathbf{\hat{H}})\mathbf{Z}_{2}\mathbf{Z}_{2}^{ \dagger}(\mathbf{H}_{\alpha\alpha}+\mathbf{\hat{H}})^{\dagger}x\right|}{ \left|\mathbf{K}_{H}\right|}\right)-\log\left(\frac{\left|\mathbf{K}_{G}+ \mathbf{G}_{2}\mathbf{Z}_{2}^{\dagger}\mathbf{G}_{2}^{\dagger}\mathbf{G}_{2}^ {\dagger}\mathbf{x}_{2}^{\dagger}\right|}{\left|\mathbf{K}_{G}\right|}\right) -\mathbf{C}_{sec,co}^{0,m}\\ g_{2}(x,y)&=\log\left(\frac{\left|(\mathbf{H}_{ \alpha\alpha}+\mathbf{\hat{H}})\mathbf{Z}_{2}\mathbf{Z}_{2}^{\dagger}( \mathbf{H}_{\alpha\alpha}+\mathbf{\hat{H}})^{\dagger}y+(\mathbf{H}_{\alpha \alpha}+\mathbf{\hat{H}})\mathbf{Z}_{2}\mathbf{Z}_{2}^{\dagger}(\mathbf{H}_{ \alpha\alpha}+\mathbf{\hat{H}})^{\dagger}x\right|}{\left|\mathbf{K}_{H} \right|}\right)-\mathrm{K}\\ g_{3}(x,y)&=x\\ g_{4}(x,y)&=y\\ h_{1}(x,y)&=x+y-P_{0,m}\end{split}  \tag{42}\]  
3.1.3 Objective Function of Power Allocation for SIMO Communication System with Active Eavesdroppers  
In this paper, we discussed the case of the passive eavesdropper. Recently, the proposed pilot spoofing attack technology masked [PERSON] to have the ability to attack [PERSON]. Therefore, we will discuss the influence of active eavesdroppers.As shown in Figure 3, in the SIMO wire-tap communication system, Alice equips with N transmit antenna, N one-antenna receivers (Bobs) equip with N one-antenna Eves. Let \(P_{N}\) denotes the pilot symbol. \(P_{N}\) is known to Eve, which concurrently send the same pilot in the training phase withe average transmission power \(P_{e}\) and [PERSON] has a perfect knowledge of \(P_{N}\). The relevant knowledge [30] of pilot frequency has been introduced in this paper and will not be repeated. Figure 3: Single-antenna eavesdroppers launch pilot spoofing attack. According to ([9]), for the LS estimator,  
\[\overline{\mathbf{h}}_{LS}=(\mathbf{K}_{B}^{H}\mathbf{K}_{B})^{-1}\mathbf{K}_{B}^ {H}z \tag{43}\]  
where \(\overline{\mathbf{h}}_{LS}\) denotes the estimate of \(\mathbf{h}\) with LS Estimator and \(\mathbf{K}_{B}^{H}\overset{\Delta}{=}\sqrt{P_{B}}A_{B}\).