One may apply the optimization algorithm finding the global minimum of the cost function to the synthesis of the shaped beam antenna pattern. Cost function may be defined as the squared magnitude of the difference between the desired beam pattern and the synthesized beam pattern with the iterating parameters such as currents and positions of antenna elements. A hybrid algorithm combining the genetic algorithm and the Levenberg-Marquardt algorithm is used to optimize the cost function of two terms, the main beam part and the weighted side lobe part, for a cosec beam pattern synthesis. Obtained cosec pattern shows that the maximum main-beam ripple of 0.5 dB and the maximum side lobe level of -35 dB are obtained for 13 elements linear array, where the side lobe weighting is 0.5 compared to the unit weight for the main beam cost function. One may decrease the maximum side lobe level to -38.5 dB but the maximum main lobe ripple is increased to 6 dB if the side lobe weighting is 2.
Another iterative method of Newton-Rapson to find the roots of the polynomial function are applied to the pattern synthesis having many prescribed nulls. One design example of linear array antenna of 14 elements shows that the maximum of 7 nulls with -200 dB may be obtained even with some nulls in the main lobe, where the initial pattern is synthesized by the Chebyschev pattern having -32 dB side lobe level. This method is also applied to the circular arrays. Here, the current amplitudes of the array elements are assumed symmetrical due to Chebyschev pattern and the phases of each element are changed. The depth of null and the deviation of the designed pattern from the initial pattern are comparable with the deterministic design methods of changing the current amplitudes and phases of array elements simultaneously.
