## 3 Beamforming of the Miniaturized L-Array
### Basic Beamforming Performance of \(L\)-Array  
The purpose of the beamformer is to maximize the antenna beam gain towards the desired direction. It is achieved by combining the signals of the different antenna elements with a proper phase shift. The basic beamforming method is to sum the sampling signals of the receiving channels after a phase compensation. In practice, this kind of phase compensation is usually carried out in the way of digital beamforming (DBF). For the L-array employed in this paper, it can be expressed as Equation (2). \[\left\{ \begin{array}{l}E(\theta,\varphi)=E_{x}(\theta,\varphi)+E_{y}(\theta, \varphi)\\ E_{x}(\theta,\varphi)=\sum\limits_{n=1}^{N_{x}}I_{x_{n}}e^{-j2n\pi d_{x}(\sin \varphi\cos\theta-\sin\varphi_{0}\cos\theta_{0})}\\ E_{y}(\theta,\varphi)=\sum\limits_{n=1}^{N_{y}}I_{y_{n}}e^{-j2n\pi d_{y}(\cos \varphi\cos\theta-\cos\varphi_{0}\cos\theta_{0})}\end{array} \right. \tag{2}\]  
where \(E(\theta,\varphi)\), \(E_{x}(\theta,\varphi)\), \(E_{y}(\theta,\varphi)\) represents the sampled values of the electric field strength of the entire array, the \(x\)-axis and \(y\)-axis's subarrays. \(N_{x}=N_{y}=3\), the number of antenna element on the two baselines. \(I_{x_{n}}\) and \(I_{y_{n}}\) are the complex sampling amplitudes of the elements distributed along the \(x\)-axis and \(y\)-axis. \((\theta_{0},\varphi_{0})\) is the basis of phase compensation, which is also the target direction of the beamforming. Naturally, \(E(\theta,\varphi)\) has the maximum value only when Equation (3) holds. Since the L-array has a 2-D structure, Equation (3) has a unique solution that enables the specific pointing at both azimuth and elevation. Unlike linear arrays, this will not cause coning ambiguity or grating lobes [19]. Once the array and the pointing direction is fixed, \(E(\theta,\varphi)\) is only determined by the incident direction \((\theta,\varphi)\) of the signal, thus it is also called the array radiation pattern. It is apparent that the basic performance of the array radiation pattern is mainly determined by the number of antennas and space between adjacent elements [20,21]. \[\left\{ \begin{array}{l}\cos\varphi\cos\theta=\cos\varphi_{0}\cos\theta_{0}\\ \sin\varphi\cos\theta=\sin\varphi_{0}\cos\theta_{0}\end{array} \right.\Rightarrow \left\{ \begin{array}{l}\varphi=\varphi_{0}\\ \theta=\theta_{0}\end{array} \right. \tag{3}\]  
The computed radiation pattern of the whole array is shown in Figure 5. The direction of the beamforming pointing is \(\varphi=113^{\circ},\ \theta=35^{\circ}\), which is the transmitting and incident direction of the oblique backscatter echoes. Figure 5a is the normalized array pattern at the frequency point of 5 MHz. Figure 5b,c are the sections of \(\varphi=113^{\circ},\ \theta=35^{\circ}\), respectively. Figure 5d-f and Figure 5g-i, respectively, are the corresponding simulation results of 10 MHz and 15 MHz. With the increase of the frequency and the relative aperture of the antenna array, the beamwidth also decreases gradually, which means better directivity. In azimuth, the beam width of the three representative frequencies are \(128^{\circ}\), \(74^{\circ}\) and \(50^{\circ}\) respectively. In the low-frequency band, such a miniaturized L-array is indeed difficult to form a very narrow beam. But in practice, even if the operating frequency is 5 MHz, it has formed an effective spatial filtering for interference signals from the other two-thirds directions. In addition, in the \(360^{\circ}\) azimuth range, except in the specified direction, there are no other grating lobes generated, which is not available for general linear array. In elevation, although the directivity is not as good as in azimuth, it has a good weakening ability for high elevation signals. The advantage of this is that the vertical echoes generated by the upward sidelobe of the transmitting antenna will be suppressed. In addition, the influence on the interpretation of the oblique backscatter signals will be reduced.