# A New Generation of Fast and Low-Memory Smart Digital/Geometrical Beamforming MIMO Antenna
## 3 Beamforming and Beamsteering  
Any arbitrary set of dipoles arranged in a uniform linear array produces a symmetrical radiation pattern on both sides of the plane on which the dipoles are placed. However, this arrangement cannot generate a single beam and steer it over the entire space surrounding the antenna. The proposed antenna [3] is a dipole antenna array that can form a single beam and steer the beam in any direction using the optimized phase delay factors. Many research studies show different types of techniques [15; 16; 17] used for optimizing the phase shifters. Among them, machine learning and artificial intelligence-based techniques are significant in the optimization process. However, these models require high computational power and memory. This paper presents a system that uses a simple mathematical model to optimize the phase delay factors. Referring to Figure 4, the radiated electric field at a particular point can be written in terms of antenna element current vectors \(I_{1}\), \(I_{2}\), \(I_{3}\),..., \(I_{n}\) for the \(N\) number of dipoles. \[E=P_{0}I_{1}e^{-j\beta r_{1}}+P_{0}I_{2}e^{-j\beta r_{2}}+\ldots+P_{0}I_{n}e^{- j\beta r_{n}} \tag{9}\]  
where \(P_{0}=Af_{tx}(\theta)\). Substituting for spherical coordinate distances \(r_{1}\), \(r_{2}\), and \(r_{n}\) by Cartesian X-Y coordinate distances, Equation (9) can be written as follows, recollecting that \(AI_{i}=Z_{i}\):  
\[E=f_{tx}(\theta) \begin{pmatrix}Z_{1}e^{j\beta(x_{1}cos\varphi+y_{1}sin\varphi) }+Z_{2}e^{j\beta(x_{2}cos\varphi+y_{2}sin\varphi)}+\ldots\\ +Z_{n}e^{j\beta(x_{n}cos\varphi+y_{n}sin\varphi)}\end{pmatrix}  \tag{10}\]  
where \(Z_{1}\), \(Z_{2}\), and \(Z_{n}\) are the complex weights that are proportional to the complex current vectors \(I_{1}\), \(I_{2}\), and \(I_{n}\), respectively. To achieve the objective of forming a resultant single beam, the values of the complex weights \(Z_{1}\), \(Z_{2}\), and \(Z_{n}\) need to be optimized such that the resultant radiated field must match a desired single beam function \(f(\varphi)\). In order to  
Figure 4: Schematic diagram of dipole placement in XY coordinate plane. optimize the phase delay factors to obtain a steerable single beam, a desired single beam function \(f(\varphi)\) is selected, as shown in Equation (11). \[f(\varphi)=Z_{1}e^{i\beta(x_{1}cos\varphi+y_{1}sin\varphi)}+Z_{2}e^{i\beta(x_{2} cos\varphi+y_{2}sin\varphi)}+\ldots+Z_{n}e^{i\beta(x_{n}cos\varphi+y_{n}sin\varphi)} \tag{11}\]  
The optimization of complex weights \(Z_{1}\), \(Z_{2}\), and \(Z_{n}\) may be achieved either analytically or iteratively [16]. Since the number of dipole elements will be limited to as few as possible, an analytical optimization method is more appropriate. Complex weights \(Z_{1}\), \(Z_{2}\), and \(Z_{n}\) can be optimized by multiplying the equation by its complex conjugates and integrating the \(\varphi\) over the limit from 0 to \(2\pi\). Here, the first term is multiplied with the \(e^{i\beta(x_{1}cos\varphi+y_{1}sin\varphi)}\) and integrated with respect to the angle \(\varphi\) over the limits from 0 to \(2\pi\). Thus,  
\[ \begin{split}\int\limits_{0}^{2\pi}f(\varphi)e^{-i\beta(x_{1} cos\varphi+y_{1}sin\varphi)}d\varphi&=Z_{1}\int\limits_{0}^{2\pi}d \varphi+Z_{2}\int\limits_{0}^{2\pi}e^{i\beta[(x_{2}cos\varphi+y_{2}sin\varphi )-(x_{1}cos\varphi+y_{1}sin\varphi)]}d\varphi\\ &+\ldots+Z_{n}\int\limits_{0}^{2\pi}e^{i\beta[(x_{n}cos\varphi+y _{n}sin\varphi)-(x_{1}cos\varphi+y_{1}sin\varphi)]}d\varphi\end{split}  \tag{12}\]  
Similarly, multiplying Equation (11) with the complex conjugates of the second term and integrating over 0 to \(2\pi\), we obtain Equation (13),  
\[ \begin{split}\int\limits_{0}^{2\pi}f(\varphi)e^{-i\beta(x_{2} cos\varphi+y_{2}sin\varphi)}d\varphi&=Z_{1}\int\limits_{0}^{2\pi}e^{i \beta[(x_{1}cos\varphi+y_{1}sin\varphi)-(x_{2}cos\varphi+y_{2}sin\varphi)]}d \varphi+Z_{2}\int\limits_{0}^{2\pi}d\varphi\\ &+\ldots+\ Z_{n}\int\limits_{0}^{2\pi}e^{i\beta[(x_{n}cos\varphi +y_{n}sin\varphi)-(x_{2}cos\varphi+y_{2}sin\varphi)]}d\varphi\end{split}