Optimizing Radiation Patterns of Thinned Arrays with Deep Nulls Fixed through Their Representation in the Schelkunoff Unit Circle and a Simulated Annealing Algorithm

Abstract

The present work develops an innovative methodology for fixing deep nulls in radiation patterns of symmetrical thinned arrays while maintaining a low side lobe level (SLL) and a high directivity, implementing an optimization strategy based on the simulated annealing algorithm (SA). This procedure optimizes a cost function that has a term for each characteristic of the desired radiation pattern and can distinguish between the deep nulls and the filled ones depending on whether they are on the Schelkunoff unit circle or not. Then, a direct extension of the methodology for planar arrays based on the separable distribution procedure is addressed. Consequently, some examples with half-wavelength spacing are presented, where the fixing of one, two, or three deep nulls in arrays of 40, 60, and 80 elements are illustrated as well as an extension to a 40 × 40-element planar array with rectangular grid and rectangular boundary, with two deep nulls fixed on each one of its main axes. Additionally, a comparison of the obtained results with a genetic algorithm (GA) alternative is performed. The main advantage of the proposed method is its ability to fix deep nulls in the radiation patterns, while maintaining an easy feeding network implementation.

Deep null fixing on optimal compromise among sum and difference patterns of thinned arrays through the Schelkunoff unit circle representation

“…This encoding is performed by normalizing every continuous value of the relative amplitudes, ranging from zero to one, and setting to bit-zero the values lower than 0.5 and to bit-one those equal or greater than 0.5. GA-based alternatives are excluded, since they present greater computational cost associated (something particularly dramatic for a high number of array elements), as demonstrated already in 21 .…”

“…The sequence of codified relative amplitudes is iteratively modified by the hybrid SA algorithm in order to find the combination whose characteristics match the desired ones in both the sum pattern (symmetric phase) and the difference pattern (anti-symmetric phase). More precisely, the procedure follows a similar strategy as the already described in 21 , but generalizing its use to compromise solutions among sum and difference far-field patterns. In such terms, a cost function is defined, where each chosen parameter of both the sum and difference radiation patterns are implemented.…”

“…To this aim, a general cost function for the process can be defined as where and are determined by particularizing a for each case, being where and are the obtained and desired SLL, respectively; and are obtained and desired normalized peak directivity on ; is the Heaviside step function ( 24 (p. 1020) ); corresponds to the number of desired nulls to be fixed; and are the obtained and desired null position by means of their polar angles and, finally, and are the different weights of the cost function. The polar angles of these null positions were obtained by introducing the Schelkunoff unit circle representation of the roots ( ) of the polynomial associated to the relative excitations, as pointed out in 21 . In such a way, the angular position of each null of the far field pattern was calculated after determining the angle of (represented in the complex plane), since …”

“…Regarding array thinning techniques and considering uniform arrays, Haupt proposed interleaved arrays by developing three different approaches (one full and two partially modes of interleaving) by means of genetic algorithms (GA) 20 . Thus, following the spirit of a very recent publication 21 where improvements on the null fixing for pencil beam patterns by means of array thinning in uniformly excited arrays were proposed, the present paper introduces an innovative technique for developing improvements on compromises between sum and difference far-field patterns.…”

“…Previous methods 10 – 20 either only study the optimization of some of the described characteristics of the pattern (SLL and directivity) or they do not use array thinning in order to simplify the feeding network. Also, none of the above-mentioned articles can fix deep, analytical nulling directions in the pattern, with the exception of 21 , which does not analyze the compromise between sum and difference patterns. Thus, the presented method overcomes state-of-the-art techniques, since it represents the first approach of addressing the optimization of SLL and directivity of thinned antenna arrays including deep null fixing in the compromise involving sum and difference patterns at the same time.…”