## 2 Related Work and System Model
### Related Work-The Artificial Noise  
Located in the null space of legitimate channel (i.e., [PERSON]'s channel), AN does not affect [PERSON]'s reception of information. For [PERSON], however, AN reduces [PERSON]'s channel capacity significantly. [PERSON] sends AN simultaneously while sending the information-bearing signal; that is,  
\[\mathbf{x}_{k}=\mathbf{w}_{k}+\mathbf{s}_{k}, \tag{3}\]  
In (3), \(\mathbf{w}_{k}\in\mathbb{C}^{N_{\text{A}}}\) denotes \(\Lambda\)N; \(\mathbf{s}_{k}\in\mathbb{C}^{N_{\text{A}}}\) denotes the information-bearing signal; and \(\mathbf{w}_{k}\) is artificial noise, which is located in the null space of \(\mathbf{H}_{k}\), such that \(\mathbf{H}_{k}\mathbf{w}_{k}=0\). Let \(\mathbf{Z}_{k}\) be a standard orthonormal basis for \(\mathbf{H}_{k}\) and \(\mathbf{v}_{k}\) be a complex random variable with the variance of \(\sigma_{v}^{2}\) such that \(\mathbf{w}_{k}=\mathbf{Z}_{k}\mathbf{v}_{k}\) and \(\mathbf{Z}_{k}^{\dagger}\mathbf{Z}_{k}=\mathbf{I}\). Then, the signals received by [PERSON] and [PERSON] are:  
\[\mathbf{z}_{k} =\mathbf{H}_{k}\mathbf{x}_{k}+\mathbf{n}_{k} \tag{4}\] \[=\mathbf{H}_{k}\mathbf{w}_{k}+\mathbf{H}_{k}\mathbf{s}_{k}+ \mathbf{n}_{k}\] \[=\mathbf{H}_{k}\mathbf{s}_{k}+\mathbf{n}_{k},\] \[\mathbf{y}_{k} =\mathbf{G}_{k}\mathbf{s}_{k}+\mathbf{G}_{k}\mathbf{w}_{k}+ \mathbf{e}_{k}, \tag{5}\]  
where \(\mathbf{y}_{k}\) is the signal received by [PERSON], and \(\mathbf{z}_{k}\) is the signal received by [PERSON]. \(\mathbf{y}_{k}\) and \(\mathbf{z}_{k}\) are Gaussian vectors. As \(\mathbf{w}_{k}\) is in the null space of \(\mathbf{H}_{k}\), we have \(\mathbf{H}_{k}\mathbf{w}_{k}=0\) and the term with \(\mathbf{w}_{k}\) vanishes in (4). That is, the artificial noise does not impact [PERSON], while [PERSON] is affected. In [4], the transmitted signal is chosen as \(\mathbf{s}_{k}=\mathbf{p}_{k}\mathbf{u}_{k}\), where \(\mathbf{u}_{k}\) is the information signal with the variance of \(\sigma_{u}^{2}\) and \(\mathbf{p}_{k}\) obeys the independent Gaussian distribution. Here, \(\mathbf{p}_{k}\) is chosen such that: (a) \(\mathbf{H}_{k}\mathbf{p}_{k}\
eq 1\), and (b) \(\|\mathbf{p}_{k}\|\)=1. In [4], [PERSON] considers two scenarios:  
1. A single-input, single-output (SISO) wireless communication system where the transmitter, the receiver, and the eavesdropper equip one antenna each, i.e., \(N_{A}=N_{B}=N_{R}=1\); and
2. A MIMO wireless communication system where the the transmitter, the receiver, and the eavesdropper each equip multiple antennas, i.e., \(N_{A}=N_{B}=N_{R}>1\). For scenario a, the variables in (4)-(6) are Gaussian complex variables. \(\log_{\text{F}}(*)\) is used to calculate entropy, so the lower bound on secrecy capacity after adding artificial noise is given by:  
\[ \begin{split} C_{sec}^{a}&=\text{I}(Z;S)-\text{I}( Y;S)\\ &=\log\left(1+\frac{|H_{k}p_{k}|^{2}\sigma_{u}^{2}}{\sigma_{u}^{2}} \right)-\log\left(1+\frac{|G_{k}p_{k}|^{2}\sigma_{u}^{2}}{E|G_{k}w_{k}|^{2}+ \sigma_{\varepsilon}^{2}}\right),\end{split}  \tag{6}\]  
where \(\text{E}|G_{k}w_{k}|^{2}=(G_{k}Z_{k}Z_{k}^{\dagger}G_{k}^{\dagger})\sigma_{k} ^{2}\). \(C_{sec}^{a}\) denotes the secrecy capacity after adding artificial noise, and \(\text{I}(\text{A};\text{B})\) denotes mutual information entropy of A and B. For scenario b, \(\mathbf{G}_{k}\) and \(\mathbf{H}_{k}\) are Gaussian complex matrices. The elements in \(\mathbf{G}_{k}\) and \(\mathbf{H}_{k}\) are Gaussian complex variables. The other variables in (4)-(6) are Gaussian vectors. It then follows that the lower bound on secrecy capacity after adding artificial noise is given by:  
\[