## 3 Beamforming of the Miniaturized L-Array
### The Weight Vector of Beamforming  
In essence, it can be understood that the essence of beamforming is to obtain a better output by weighting and summing the antennas' signal. Once \(E(\theta,\varphi)\) is rewritten as Equation (4), the weights can be considered as the phase compensations required in the beamforming. \[E(\theta,\varphi)=\sum_{n=1}^{N_{x}}I_{x_{n}}e^{-j2 mrd_{x}(\sin \varphi\cos\theta)/\lambda}e^{j2 mrd_{x}\sin\varphi\cos\theta_{0}/\lambda}+ \sum_{n=N_{x}+1}^{N_{x}+N_{y}}I_{y_{n}}e^{-j2(n-N_{x})mrd_{y}(\cos\varphi\cos \theta)/\lambda}e^{j2(n-N_{x})mrd_{y}\cos\varphi\cos\theta_{0}/\lambda}\] \[=w^{H}s=\sum_{n=1}^{N_{x}+N_{y}}m_{x}^{e}s_{n} \tag{4}\] \[w=[w_{1},w_{2},\cdots w_{n}]^{T},w_{n}=\left\{ \begin{array}{l}e^ {j2 mrd_{x}\sin\cos\theta_{0}/\lambda},1\leq n\leq N_{x}\\ e^{j2(n-N_{x})mrd_{y}\cos\varphi\cos\theta_{0}/\lambda},N_{x}<n\leq N_{x}+N_{y }\\ \end{array} \right.\] \[s=[s_{1},s_{2},\cdots s_{n}]^{T},s_{n}=\left\{ \begin{array}{l}I _{x_{n}}e^{-j2 mrd_{x}\sin\varphi\cos\theta/\lambda},1\leq n\leq N_{x}\\ I_{y_{n}}e^{-j2(n-N_{x})mrd_{y}\cos\varphi\cos\theta/\lambda},N_{x}<n\leq N_{x}+N _{y}\end{array} \right.\]  
where \(H\) indicates the conjugate transposition, \(\ast\) represents the complex conjugation and \(T\) is the transpose symbol. \(I\) represents the integrated signal arriving at the antenna. The signal sequence and the weight vector are \(w\) and \(S\) respectively. If there are \(N\) snapshots of the signal, the output of the beamformer for time \(t\) can be expressed as \(y(t)\) in Equation (5). \[y(t)=w^{H}s(t),t=1,2,\cdots,N \tag{5}\]  
Figure 5: (**a**) is the normalized array pattern at the frequency point of 5 MHz. (**b**,**c**) are the sections of \(\varphi=113^{\circ},\ \theta=35^{\circ}\), respectively. (**d**–i**) are the corresponding simulation results of 10 MHz and 15 MHz. For a particular signal sequence \(d(t)\) of the incident direction of \((\theta_{d},\varphi_{d})\), reference to Equation (1), \(s(t)\) can be expressed as Equation (6). \[s(t)=a(\theta_{d},\varphi_{d})d(t) \tag{6}\]  
Thus, the average power can be expressed as Equation (7). \[P(w)=E[\big{|}y(t)\big{|}^{2}]=w^{H}E[s(t)s^{H}(t)]w=w^{H}Rw=E[\big{|}d(t) \big{|}^{2}]\big{|}w^{H}a(\theta_{d},\varphi_{d})\big{|}^{2} \tag{7}\]  
where \(E[\bullet]\) indicates the expectation, and \(R\) is the autocorrelation matrix of the echo signals of the array. Combining Equations (1) and (4), Equation (8) can be obtained. It is clear that in the ideal situation without interference signal and noise, the maximal output of the array is \(N_{x}+N_{y}\) times compared with a single antenna. \(w\) is the optimal weight vector of the beamforming. \[\big{|}w^{H}a(\theta_{d},\varphi_{d})\big{|}^{2}=N_{x}+N_{y} \tag{8}\]  
However, when there are noises and interference signals from other directions, \(s(t)\) and \(P(w)\) should be rewritten into Equations (9) and (10). \[s(t)=[a(\theta_{d},\varphi_{d}),a(\theta_{i_{1}},\varphi_{i_{1}}),\cdots,a( \theta_{i_{J}},\varphi_{i_{J}})]\left[ \begin{array}{c}d(t)\\ i_{1}(t)\\ \vdots\\ i_{J}(t)\end{array} \right]+\left[ \begin{array}{c}n_{0}(t)\\ n_{1}(t)\\ \vdots\\ n_{J}(t)\end{array} \right] \tag{9}\]  
\[P(w)=E[\big{|}d(t)\big{|}^{2}]\big{|}w^{H}a(\theta_{d},\varphi_{d})\big{|}^{2 }+\sum_{k=1}^{J}E[\big{|}i_{k}(t)\big{|}^{2}]\big{|}w^{H}a(\theta_{i_{k}}, \varphi_{i_{k}})\big{|}^{2}+\|w\|^{2}\sum_{k=1}^{J}\|n_{k}(t)\|^{2} \tag{10}\]  
where \(i_{k}(t)\) is the \(k\)th interference signal with the incident direction of \((\theta_{i_{k}},\varphi_{i_{k}})\). The corresponding array manifold is \(a(\theta_{i_{k}},\varphi_{i_{k}})\). \(J\) is the number of interference signals. In addition, \(\|\bullet\|\) represents the norm. In this case, the weight vector \(w\) is no longer optimal. Although the power of the desired signal is still maintained, due to the interaction of interference signals and noises, the output of the whole array may not have the best signal to interference noise ratio (SINR). For the large arrays, this problem is not serious.