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
N
-element array it is shown that, when
M
single nulls are imposed on a given "quiescent" pattern, the optimum solution for the constrained pattern is the initial pattern and a set of
M
-weighted
(\sin Nx)/\sin x
-beams. Each beam is centered exactly at the corresponding pattern null, irrespective of its relative location. For the case of higher order nulls, the
n
th pattern derivative is similarly canceled by the
n
th derivative of a
(\sin Nx)/\sin x
-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.