# An Overview of Key Technologies in Physical Layer Security
## 3 Secure Multi-Antenna Techniques
### Beamforming
 \tag{21}\]  
The solution to the beamforming problem provides the optimal vector \(\mathbf{w}_{i}\), which maximizes the SNR of the desired user. This results in focusing the beam in one direction, and the process is referred to as electronic steering. One beamforming vector is assigned to each legitimate user and is matched to their channel. The beamforming design problem in PLS has been well-investigated in many studies in the literature, which can be found in [58; 59; 60; 61; 62; 63; 64], with the aim of developing algorithms that minimize the interference and also maximize secrecy of transmission. We summarize some of the results from these studies here. One approach which was considered in [58] involves using semi-definite relaxation to obtain the optimal beamforming solution, which minimizes the transmission power subject to SINR constraints. In the paper, they showed that the quadratic optimization problems with non-convex and discontinuous constraints could be recast as SDP with additional constraints, which imposes that the solution matrices must be of rank one. Another study employing semi-definite relaxation was proposed in [59], wherein they used Taylor expansion to solve the optimization problem. The author's study proved that their proposed algorithm outperformed both the signal-to-leakage-and-noise ratio (SLNR)-based algorithm and zero-forcing beamforming. The SLNR algorithm and zero-forcing were employed to minimize the power leaking to the channels of other users. Another novel approach of path-following algorithm was proposed in [60; 61], which used a simple quadratic program to perform iterations for finding the optimal transmit beamformers. The QoS, which is given in terms of both the user's secrecy throughput and the network secure energy efficiency, is optimized through the use of the obtained beamformers. The algorithm proposed in [60; 61] offers a better performance when compared with the existing methods based on zero-forcing beamformers. Authors in [51; 51] further studied the algorithm proposed by [60; 61] using a different approach. The authors defined secrecy throughput in terms of outage probability. It was found that this approach offered a more practical beamforming design solution. Interference alignment (IA) was proposed by [65; 66] as an excellent solution for interference management in multi-user wireless networks to significantly improve the sum-rate. The concept of the IA technique is that the transmitted signals are directed to concentrate the interference in the particular sub-spaces at the unintended receivers, thus opening up interference-free sub-spaces to transmit the desired signal to the intended user. Following the proposal, several research works in [64; 67; 68; 69] have been conducted to develop IA algorithms for directing the interference in a manner that is detrimental to the illegitimate receiver while ensuring that the legitimate receiver is not severely affected. To enhance the desired signal gain and suppress the undesired interference and the noise signal, authors in [63] employed spatial degrees of freedom.