## 3 Proposed Solution  
Figure 2 presents a flow chart illustrating the methodology employed for synthesizing the radiation field pattern in the deformed structures of conformal array. Figure 2: Proposed compensation technique for the radiation pattern correction. The electric field pattern is used as the objective function, and the problem is reformulated as a constrained convex optimization problem. The use of constraint convex optimization enables to reach the most optimum performance of the antenna system by finding the best amplitude and phase values for the antenna array. To transform the radiation pattern synthesis problem into a convex optimization problem, the objective function and the constraints are expressed in a form that is agreeable to convex optimization techniques. This involves the use of linear and quadratic programming. The objective of the proposed algorithm is to determine the excitation amplitudes and phases for the antenna elements that produce the desired radiation pattern. The correction of the radiation field pattern synthesis involves adjusting the weights of an existing antenna array to correct for specific deformation from a desired radiation pattern. Convex optimization is used to solve this problem, and several constraint vectors are adopted to ensure that the optimization problem is tractable and can be efficiently solved. The convex optimization itself contains linear constraints that can be formulated to reduce the search space. However, using this approach, the compensated pattern will perform well at the mainlobe peak only and null points but does not care for the rest of the radiation pattern. As a result, mainlobe beamwidth distortion, higher side-lobe level at the edges, and all areas of field away from the constraint points can be distorted. The error performance of this approach, however, tends to improve as more constraints are strategically placed across the radiation pattern, especially at extremal points. Nevertheless, it is worth noting that the maximum number of constraint points has an upper limit. Therefore, convex optimization is introduced within the loops after constraining the pattern in order to allow the corrected/measured field to track the desired field pattern. This closely ensures that the desired null's location, direction of main beam, and sidelobe levels are accurately achieved. These constraint vectors are as follows:  
* Unit-modulus constraint: The weights of the antenna elements have a magnitude of one over maximum value of mainlobe, as this ensures that the power radiated by the array is conserved. The unit-modulus constraint is formulated as follows: \[||w_{m,n}\,||^{2}=1\] (4) where \(||w_{m,n}\,||^{2}\) denotes the squared L2-norm and is non-negative. * Nulling constraint: The field pattern of the antenna array is taken as constrained to have nulls in certain directions, such as to minimize interference from unwanted signals. Mathematically, \[ \begin{array}{l}\left|w_{m,n}{}^{H}\right.\ast\left.\alpha_{1}(\theta,\ \varphi)\right|^{2}=0\\ w_{m,n}{}^{H}\right.\ast\left.\alpha_{2}(\theta,\ \varphi)\right|^{2}=0\\ \cdots\cdots\\ \left|w_{m,n}{}^{H}\right.\ast\left.\alpha_{k}(\theta,\ \varphi)\right|^{2}=0\end{array} \] (5) where \(w_{m,n}{}^{H}\) denotes the Hermitian transpose of the weight vector \(w_{m,n}\),
* denotes complex conjugation, \(\alpha_{i}(\theta,\ \varphi)\) denotes the steering vector for the specific direction along \(z\)-axis, and the equality constraints ensure that the radiation pattern of the antenna array has nulls in the desired directions. * Constant-envelope constraint: The envelope of the array factor is constant. The constant-envelope constraints ensure that the power radiated by the array is constant over all angles. * Directional constraint: The radiation pattern such as a specific beamwidth or sidelobe level of the antenna array is taken as constrained along \(z\)-axis. * Hermitian symmetry constraint: The array factor is symmetric about the zero-frequency axis.