## 3 Beamforming Processing
### Beam Shaping  
The concept of beam shaping is vast and brings together all the techniques that, through feeding of the array, allows the modification of its radiation shape. Typically, this concept is used in communications to mitigate the influence of interfering signals through side lobes (reducing or eliminating them) or confining the width of the main lobe.  
There are different reported methods in the literature, such as the Schelkunoff polynomial method, minimum mean-square error (MMSE) weight and many others, and in this work, two methods, Dolph-Tschebyscheff and binomial [27], were applied in the proposed digital system and tested to demonstrate their impact on the radiation of the antenna array when compared to the uniform array.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Steering Angle \(\theta\) (\({}^{\circ}\))** & \(\alpha_{1}\) (\({}^{\circ}\)) & \(\alpha_{2}\) (\({}^{\circ}\)) & \(\alpha_{3}\) (\({}^{\circ}\)) & \(\alpha_{4}\) (\({}^{\circ}\)) \\ \hline \(40^{\circ  
Using the theoretical expressions of [27], for the array of N = 4 elements, the feeding distributions were estimated for the binomial method and for the Dolph-Tschebyscheff technique using different values of side lobe level (SLL), such as \(-15\) dB, \(-20\) dB and \(-25\) dB. The SLL represents the signals being radiated in the unwanted direction. Applying such methods in practical application could be very helpful to reduce the interferences.  
These distributions are provided in Table 2 and were applied to the theoretical Equation (1). In the Matlab, the different radiation patterns, considering all the techniques, were calculated using (1) and multiplying by the theoretical beam pattern of a microstrip patch antenna (element factor), which is the radiating element chosen. The results are shown in Figure 11.  
According to Figure 11, it is possible to observe several aspects resulting from the use of each method. First, using the uniform distribution, with all elements fed with equal amplitude and phase, a higher side lobe is observable, and the amplitude of this lobe reduces with the application of the different methods, namely using the Dolph-Tschebyscheff technique (for the different SLL estimations), and in particular using the binomial method, for which the side lobes even disappear. It can also be proved that the amplitude distribution from the application of the Dolph-Tschebyscheff method for very small SLL values will tend towards the binomial distribution.  
Another important aspect that can be taken from Figure 11 is the variation of the half power beam width (HPBW), using the different methods, observing that this value increases as the value of the SLL decreases.  
The standard definition of beam width is the angle aperture from which most of the power is radiated from antennas [27]. It is a very important characteristic of an antenna array, and a  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Weight Method** & **a1** & **a2** & **a3** & **a4** \\ \hline Uniform & 1 & 1 & 1 & 1 \\ Binomial & 1 & 3 & 3 & 1 \\ Dolph–Tschebyscheff (\(-15\) dB) & 1 & 1.33 & 1.33 & 1 \\ Dolph–Tschebyscheff (\(-20\) dB) & 1 & 1.74 & 1.74 & 1 \\ Dolph–Tschebyscheff (\(-25\) dB) & 1 & 2 & 2 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Estimated amplitude distribution.  
Figure 11: Theoretical radiation patterns considering the distribution weights.  
crucial characteristic in phased array antennas and beamforming, because of the useful advantage of minimizing unwanted interference signals by controlling the main lobe characteristics, beam width and the side lobes.  
The concept of half power beam width (HPBW) is commonly used and reflects the angular aperture in which the gain of the antenna falls 3 dB. There are theoretical expressions reported in the literature [27] that estimate this value.