# An Overview of Key Technologies in Physical Layer Security
## 3 Secure Multi-Antenna Techniques
### Beamforming  
Beamforming is a signal processing technique that is used to transmit signals effectively in intended directions to give a maximum signal difference between the receiver in the intended direction and the one in the unintended direction. Beamforming forms a beam in the direction of the desired recipient to maximize the signal-to-noise power ratio while suppressing the reception or transmission in the direction of the unintended user, Figure 11. This significantly improves the energy efficiency of the system because the energy is transmitted or focused in a particular direction rather than being spread out in a diffused fashion. Figure 10: Secure multi-antenna techniques. Beamforming can be used at both the transmitting and receiving ends to achieve spatial selectivity, i.e., transmit beamforming and receive beamforming. Transmit beamforming steers the transmitted signal towards the intended receiver by finding the best possible channel among all the transmit antennas. Beamforming is one of the key techniques in PLS and has been widely studied in the literature [49; 52; 53; 54; 55; 56; 57]. A beamforming problem in PLS involves steering the transmitted signal towards the desired user while taking into account an interfering user trying to decode the transmitted information, Figure 11. To demonstrate the beamforming optimization problem in PLS we consider a MISO system shown in Figure 12 in which the transmitter uses transmit beamforming to communicate with \(K\) users. We assume that the transmitter is equipped with \(N_{t}\) transmit antennas, and the legitimate user and the eavesdropper each have a single receiving antenna. Therefore, the received signal at legitimate user \(i\) and their equivalent \(\mathtt{SINR}\) are given by  
\[ \begin{split} y_{i}=\mathbf{h}_{i}^{H}\mathbf{x}_{i}+\sum \limits_{\begin{subarray}{c}k=1\\ k\
eq i\end{subarray}}^{K}\mathbf{h}_{i}^{H}\mathbf{x}_{k}+n_{i},\\ \mathtt{SINR}=\frac{|\mathbf{h}_{i}^{H}\mathbf{w}_{i}|^{2}}{| \sum\limits_{\begin{subarray}{c}k=1\\ k\
eq i\end{subarray}}^{K}\mathbf{h}_{i}^{H}\mathbf{w}_{k}|^{2}+\sigma^{2}} \end{split}  \tag{19}\]  
Figure 11: Beamforming. Figure 12: System model of multi-user and eavesdropper pairs with beamforming. Similarly, the signal received signal by the \(i^{th}\) eavesdropper and their equivalent \(\mathtt{SINR}\) can be given by  
\[ \begin{split} z_{i}=\mathbf{g}_{i}^{H}\mathbf{x}_{i}+\sum\limits_{ \begin{subarray}{c}k=1\\ k\
eq i\end{subarray}}^{K}\mathbf{g}_{i}^{H}\mathbf{x}_{k}+m_{i},\\ \mathtt{SINR}=\frac{|\mathbf{g}_{i}^{H}\mathbf{w}_{i}|^{2}}{\| \sum\limits_{\begin{subarray}{c}k=1\\ k\
eq i\end{subarray}}^{K}\mathbf{g}_{i}^{H}\mathbf{w}_{k}|^{2}+\sigma^{2}} \end{split}  \tag{20}\]  
where \(\mathbf{x}_{i}\in\mathbb{C}^{N_{T}\times 1}\) is the transmitted signal symbol of the desired user \(i\) with corresponding beamforming vector \(\mathbf{w}_{i}\in\mathbb{C}^{N_{T}\times 1}\), \(\mathbf{h}\in\mathbb{C}^{N_{T}\times 1}\) and \(\mathbf{g}\in\mathbb{C}^{N_{T}\times 1}\) are the channel vectors of the desired user and eavesdropper respectively, and \(n_{i}\) and \(m_{i}\) are the corresponding AWG noise vectors for the \(i^{th}\) user and eavesdropper with zero mean and noise power \(\sigma^{2}\). The covariance matrix of the transmitted signal is given by \(\mathbf{R}_{n}=\mathbb{E}\{\mathbf{x}_{i}\mathbf{x}_{i}^{H}\}\). The objective problem of a typical beamforming design scheme is to minimize the interference signal at the desired user so that they receive the transmitted signal with the desired QoS, which is usually described by the constraint of the \(\mathtt{SINR}\) greater or equal to the given threshold amount of the \(i^{th}\) user, i.e., \(\mathtt{SINR}\geq\rho_{i}\). Therefore, the beamforming optimization problem can be written as follows:  
\[ \begin{split}\underset{\mathbf{w}_{i}}{\text{min}}& |\sum\limits_{\begin{subarray}{c}k=1\\ k\
eq i\end{subarray}}^{K}\mathbf{h}_{i}^{H}\mathbf{w}_{k}|^{2}+\sigma^{2}\\ \text{s.t.}&\mathtt{SINR}\geq\rho_{i}\\ &||\mathbf{w}_{i}||^{2}\leq P_{T}\end{split}