## 3 Beamforming of the Miniaturized L-Array
### Adaptive Beamforming
#### 3.3.2 Eigenspace-Based Beamformer  
Eigenspace-based beamforming (ESB) algorithm can be regard as an improvement of SMI. Assuming that there is one desired signal but with \(J\) interferences (which is generally assumed that \(J<N_{x}+N_{y}\)), the eigenvalue decomposition of the covariance matrix \(R\) with infinite sampling points can be computed as Equation (14).  
\[R=\sum_{i=1}^{l+1}\lambda_{i}u_{i}u_{i}^{H}+\sigma_{n}^{2}\sum_{i=l+2}^{N_{x}+ N_{y}}u_{i}u_{i}^{H} \tag{14}\]  
where \(\lambda_{1}\geq\lambda_{2}\geq\cdots\lambda_{l+1}>\lambda_{l+2}=\cdots= \lambda_{N_{x}+N_{y}}=\sigma_{n}^{2}\) are the \(N_{x}+N_{y}\) eigenvalues, \(u_{i}(i=1,2,\cdots,N_{x}+N_{y})\) are the corresponding eigenvectors. Then, in SMI, the optimal weight vector \(w_{opt}\) can be rewritten as Equation (15):  
\[w_{opt}=\mu E_{s}D_{s}^{-1}E_{s}^{H}a(\theta_{d},\varphi_{d})+\mu E_{n}D_{n}^{ -1}E_{n}^{H}a(\theta_{d},\varphi_{d}) \tag{15}\]  
where \(D_{s}=diag(\lambda_{1},\cdots,\lambda_{l+1})\), \(D_{n}=diag(\lambda_{l+2},\cdots,\lambda_{N_{x}+N_{y}})\). \(E_{s}=[u_{1},\cdots,u_{l+1}]\) and \(E_{n}=[u_{l+2},\cdots,u_{N_{x}+N_{y}}]\) constitute the signal subspace and noise subspace respectively. Under the ideal condition, Equation (16) should be established [29]. Thus, the weight vector \(w_{opt}=w_{0}=\mu E_{s}D_{s}^{-1}E_{s}^{H}a(\theta_{d},\varphi_{d})\) should only have the signal subspace component.  
\[E_{n}^{H}a(\theta_{d},\varphi_{d})=0 \tag{16}\]  
However, considering the actual situation, the number of snapshots is limited. The signal will leak into the noise subspace. Thus, Equation (16) is no longer established, which will lead to the decline of the output SINR and may cause the pattern distortion [30]. This phenomenon is more serious in the case of high SNR.  
An improved method proposed by [PERSON] et al. is to discard the components of the weight vector in the noise subspace. That is, the weight vector is projected to the signal subspace. Then the weight vector can be rewritten as Equation (17). As the weight vector is located in the signal subspace, the weakening of the signal caused by signal subspace leakage of SMI algorithm can be significantly reduced [31].  
\[w_{1}=\mu E_{s}D_{s}^{-1}E_{s}^{H}a(\theta_{d},\varphi_{d})=E_{s}E_{s}^{H}w_{0} \tag{17}\]  
Figure 7 shows the simulation results of SMI and ESB beamformer when the SNR is adjusted to 15 dB. Figure 7a is the array pattern of SMI beamformer, Figure 7b,c respectively show the sections of \(\varphi=113^{\circ}\) and \(\theta=35^{\circ}\). In the case of high SNR, the SMI pattern is distorted and the beam pointing is deflected. As shown in Figure 7b,c, the array gain of the desired direction is weakened and another high sidelobe is generated. In contrast, the array pattern of ESB beamformer shown in Figure 7d still has a good performance on the L-array. The beam pointing is still correct. In the sections of \(\varphi=270^{\circ}\), \(\theta=80^{\circ}\) showing in Figure 7e,f, the interference direction forms a deeper depression.  
However, the ESB also has its disadvantages. When the desired signal is too weak, discarding the noise subspace will cause a large error. Therefore, ESB has a better performance for stronger desired signal [26]. This is also verified by the processing of actual signals below.