Synthesis of antenna patterns with prescribed nulls

The least mean square pattern synthesis method is extended to include constraints such as pattern nulls or pattern-derivative nulls at a given set of angles. The problem is formulated as a constrained approximation problem which is solved exactly, and a clear geometrical interpretation of the solution in a multidimensional vector space is given. The relation of the present method to those of constrained gain maximization and signal-to-noise ratio (SNR) maximization is discussed and conditions for their equivalence stated. For a linear uniform -element array it is shown that, when single nulls are imposed on a given "quiescent" pattern, the optimum solution for the constrained pattern is the initial pattern and a set of -weighted -beams. Each beam is centered exactly at the corresponding pattern null, irrespective of its relative location. For the case of higher order nulls, the th pattern derivative is similarly canceled by the th derivative of a -beam. In addition, simple quantitative expressions are derived for the pattern change and gain cost associated with the forced pattern nulls. Several illustrative examples are included.