# Phased Antenna-Array Synthesis Using Taylor-Series Expansion and Neural Networks
## 2 Modern Radar and Phased Antenna Arrays  
Phased-array radars use antenna arrays that can be electronically controlled to steer the radar beam without mechanical movement, enabling fast scanning and efficient target detection and tracking. The array factor (\(AF\)) plays a crucial role in beamforming and is defined as the sum of contributions from each antenna element in the array, where each element is fed with a phase-adjusted signal [22; 23]. For a linear array of \(N\) elements spaced with \(d\), the array factor is given by the following:  
\[AF(\theta)=\sum_{n=0}^{N-1}e^{j(nkd\cos\theta+\phi_{n})} \tag{1}\]  
where \(k=\frac{2\pi}{\lambda}\) is the wave number, \(\lambda\) is the wavelength of the radar signal, \(\theta\) is the observation angle, and \(\phi_{n}\) is the phase shift applied to each antenna element. In the case of a linear phase shift with an angle, \(\theta_{0}\), of the main beam direction, each element receives a phase, \(\phi_{n}=-nkd\cos\theta_{0}\), where \(\theta_{0}\) is the direction of the main beam. This allows the beam to be steered towards \(\theta_{0}\) without requiring the physical movement of the antennas [24; 25]. The array factor is often expressed in a closed form as follows:  
\[AF(\theta)=\frac{1-e^{jN\psi}}{1-e^{j\psi}} \tag{2}\] \begin{table}
\begin{tabular}{c p{142.3 pt} p{142.3 pt} p{142.3 pt}} \hline \hline
**Method** & **Description** & **Advantages** & **Disadvantages** \\ \hline Taylor [18] & Taylor coefficients adjust antenna phases. & Easy to implement & Requires spatial distribution knowledge. \\ \hline Dolph-Chebyshev [19] & Minimizes amplitude error using Chebyshev polynomials. & High sidelobe attenuation. & Complex for certain antenna configurations. \\ \hline PSO [20] & Uses particle swarm optimization for parameter adjustment. & Quick convergence. & Convergence depends on initial setup. \\ \hline GA [21] & Utilizes genetic algorithms for parameter optimization. & Suitable for complex designs. & Requires significant computation times. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Beamforming methods comparison. where \(\psi=kd(\cos\theta-\cos\theta_{0})\). By manipulating this equation, the array factor can also be expressed in a trigonometric form:  
\[AF(\theta)=e^{i\frac{(N-1)}{2}\psi}\frac{\sin\left(\frac{N}{2}\psi\right)}{\sin \left(\frac{1}{2}\psi\right)} \tag{3}\]  
This formulation shows that the main beam is directed toward \(\theta_{0}\), and the radiation pattern, including the main lobe width and side lobes, depends on the number of elements, \(N\), the element spacing, \(d\), and the phase shift, \(\beta\). The angular resolution of the radar, which defines the ability to distinguish closely spaced targets, is determined by the width of the main lobe at \(-3\) dB, calculated as follows:  
\[\theta_{HPBW}\approx\frac{0.886\lambda}{Nd\cos\theta_{0}} \tag{4}\]  
Thus, increasing the number of elements in the array results in a narrower beam and better angular resolution, but it also requires optimal element spacing to avoid the formation of grating lobes. If the element spacing, \(d\), exceeds \(\lambda/2\), unwanted side lobes, or grating lobes, may appear, leading to detection errors. To mitigate these effects, windowing functions such as Taylor, Chebyshev, or Hamming windows are applied to reduce side lobes, optimizing radar performance without significantly increasing the main beam width [26; 27]. Among these, Taylor-series expansion is favored in this work due to its analytical tractability and ability to produce physically interpretable excitation profiles with controlled sidelobe behavior. Unlike Chebyshev- or Fourier-based designs, Taylor excitation enables the smooth tapering of the amplitude distribution, which integrates naturally with neural network optimization and facilitates convergence. This balance between simplicity, control, and compatibility with data-driven tuning makes Taylor series a strong candidate for hybrid synthesis frameworks.