Extension of Orchard's pattern synthesis technique for overdetermined systems

The problem of synthesizing antenna patterns with controlled nulls has been extensively studied in the literature. Prescribed nulls in the radiation pattern can be used to suppress interferences from specific directions. The need for a broad antenna null often arises when the direction of arrival of the unwanted interference may vary slightly with time or may not be known exactly, and where a comparatively sharp null would require continuous steering for obtaining a reasonable value for the signal-to-noise ratio. The SVD algorithm is useful in solving under and overdetermined systems of linear equations. This method is used to solve an overdetermined system obtained from that of the Orchard (1985) method (of linear array pattern synthesis) wherein several extra equations are introduced to obtain patterns with fixed nulls and to control the width of some angular regions of interest. In order to test the precision of this proposed method, comparisons with other optimization methods have been carried out. A cost function was built, as the squared difference between the desired and obtained pattern, as a function of the root displacements. The Fletcher-Powell (variable metric) and the Polak-Ribiere (conjugate gradient) methods were used to optimize the cost function.