# Phased Antenna-Array Synthesis Using Taylor-Series Expansion and Neural Networks
## 3 The Taylor Method  
The simulated antenna consists of 16 identical microstrip patch elements arranged in a uniform linear array with \(\lambda/2\) spacing. This structure serves as the physical basis for the optimization framework and simulation results presented in Figures 2-5. The Taylor method is a classical technique used in antenna array design to control how energy is distributed across the radiation pattern. It enables designers to fine-tune parameters such as beamwidth and sidelobe level by adjusting excitation amplitudes across the array elements using a mathematical series expansion. When a precise expression of the electromagnetic field is not easily available, Taylor expansion allows engineers to locally approximate the field distribution and apply optimized amplitude weights to shape the desired beam [25; 33; 34; 35]. Using spherical coordinates \((r,\theta,\phi)\), where \(r\) represents the radial distance from the observation point, \(\theta\) represents the azimuthal angle, and \(\phi\) represents the elevation angle, the Taylor method can be extended to approximate the spatial distribution of the electromagnetic field as follows [36; 37]:  
\[E(r,\theta,\phi)\approx E(r_{0},\theta_{0},\phi_{0})+\left.\frac{\partial E}{ \partial\theta}\right|_{(r_{0},\theta_{0},\phi_{0})}(\theta-r_{0})+\left.\frac {\partial E}{\partial\theta}\right|_{(r_{0},\theta_{0},\phi_{0})}(\phi- \phi_{0})+\left.\frac{\partial E}{\partial\phi}\right|_{(r_{0},\theta_{0}, \phi_{0})}(\phi-\phi_{0}) \tag{5}\]  
Figure 1: General arrangement of a linear antenna array [32]. In this expression, the following applies:  
* \(E(r,\theta,\phi)\) represents the spatial distribution of the electromagnetic field of the antenna array. * \((r_{0},\theta_{0},\phi_{0})\) is the reference point around which the Taylor series is developed. * \(\frac{\partial E}{\partial r},\frac{\partial E}{\partial\theta}\), and \(\frac{\partial E}{\partial\phi}\) are the partial derivatives of \(E\) with respect to \(r\), \(\theta\), and \(\phi\) respectively, evaluated at the reference point \((r_{0},\theta_{0},\phi_{0})\). Figure 3: [PERSON] antenna array with Kaiser window \(N=16\), \(d=\frac{\lambda}{7},\theta 60^{\circ}\). Figure 2: Synthesis of antenna weights for different lobe widths and array sizes. The Taylor method is utilized in antenna design to dictate the amplitude attenuation of elements within an antenna array, often characterized by a single parameter such as the main lobe width. This amplitude attenuation, when applied to the array elements, directly impacts the array factor, a mathematical function delineating each array element's contribution to the overall radiation pattern formation. Thus, employing the [PERSON] method to ascertain the amplitude of array elements directly influences the array factor and subsequently the comprehensive radiation pattern of the antenna array [38; 39; 40]. Through parameter adjustments in the Taylor method, like modifying the main lobe width, one can manipulate the amplitude of the array factor, enabling precise control and optimization of the antenna array's radiation pattern to align with specific requirements. This Algorithm 1 outlines the steps to synthesize a Taylor-Kaiser antenna array. It starts by choosing the appropriate window and calculating coefficients based on the given specifications. Then, for each antenna element, it calculates the weight and position. If the  
Figure 4: Taylor antenna array with Kaiser window \(N=16\), \(d=\frac{\lambda}{2}\), @90\({}^{\circ}\). performance is satisfactory, it finalizes the design; otherwise, it adjusts the parameters to improve performance. ```
1:functionSynthesizeAntenna(specs)
2:Step 0:InitializeParameters
3: Define the number of elements \(N\) and the desired sidelobe level (SLL). 4: Set the beamwidth and aperture size according to specs. 5:Step 1:SelectWindow Function
6: Choose the appropriate window (Taylor or Kaiser) based on SLL and beamwidth.